Regularity properties of k-Brjuno and Wilton functions

Abstract

We study functions related to the classical Brjuno function, namely k-Brjuno functions and the Wilton function. Both appear in the study of boundary regularity properties of (quasi) modular forms and their integrals. We consider various possible versions of them, based on the $\alpha$-continued fraction developments. We study their BMO regularity properties and their behaviour near rational numbers of their finite truncations. We then complexify the functional equations which they fulfill and we construct analytic extensions of the k-Brjuno and Wilton functions to the upper half-plane. We study their boundary behaviour using an extension of the continued fraction algorithm to the complex plane. We also prove that the harmonic conjugate of the real k-Brjuno function is continuous at all irrational numbers and has a decreasing jump of $\pi /q^k$ at rational points p/q.

1 Introduction

Let $G$ denote the Gauss map defined on the interval [0, 1) by $G(0)=0$ and $G(x)=\left\{ \frac{1}{x}\right\}$ otherwise, where $\{x\}=x-\lfloor x\rfloor$ and let

$$\begin{aligned} \beta _j(x) = \prod _{i=0}^j G^i(x) \end{aligned}$$

for $j\ge 0$ with the convention $\beta _{-1}(x) = 1$, where $G^j$ denotes the jth iteration of G, not the power of G.

1.1 Brjuno function

In 1988, Yoccoz introduced the following function - now called Brjuno function - defined for irrational numbers $x\in [0,1]\setminus \mathbb {Q}$ as

$$\begin{aligned} B_1(x)=\sum _{n=0}^{\infty }\beta _{n-1}(x)\, \log \left( \frac{1}{G^n(x)}\right) , \end{aligned}$$

see [25, 27, 37], see Fig. 1 for its graph.

Let $\frac{p_n(x)}{q_n(x)}$ denote the nth convergent of x with respect to its continued fraction expansion. The series $B_1(x)$ converges if and only if

$$\begin{aligned} \sum _{n=0}^{\infty } \frac{\log (q_{n+1}(x))}{q_n(x)} < \infty . \end{aligned}$$

This condition is called Brjuno condition and was introduced by Brjuno in the study of certain problems in dynamical systems, see [8, 9]. The points of convergence are called Brjuno numbers. The importance of Brjuno numbers comes from the study of analytic small divisors problems in dimension one. Indeed, extending previous fundamental work by Siegel [34], Brjuno proved that all germs of holomorphic diffeomorphisms of one complex variable with an indifferent fixed point with linear part $e^{2\pi i x}$ are linearisable if x is a Brjuno number. Conversely, in 1988 Yoccoz [37, 38] proved that this condition is also necessary. Similar results hold for the local conjugacy of analytic diffeomorphisms of the circle [39] and for some complex area-preserving maps [11, 23]. Moreover, the sum of the Brjuno function and the logarithm of the radius of convergence of the quadratic polynomial has a continuous extension to the real line [2] and this interpolation is conjectured to be Hölder continuous with exponent 1/2 [25]. This has been recently proved for high-type irrational numbers [10]. The Brjuno condition has been of interest in various contexts. For instance, it is conjectured that it is optimal for the existence of real analytic invariant circles in the standard family [20,21,22, 28]. See also [4, 15] and the references therein for related results.

Furthermore, the Brjuno function is $\mathbb {Z}$–periodic and satisfies the functional equation

$$\begin{aligned} B_1(x)=-\log (x)+xB_1\left( \frac{1}{x}\right) \end{aligned}$$

(1.1)

for $x\in (0,1)$. The second author together with Moussa and Yoccoz investigated the regularity properties of $B_1$ in [25] and later constructed an analytic extension of $B_1$ to the upper half-plane $\mathbb {H}= \{z\in \mathbb {C}:\text {Im}z>0\}$ [26]. Let T denote the linear operator

$$\begin{aligned} Tf(x)=xf\Big (\frac{1}{x}\Big ) \end{aligned}$$

(1.2)

acting, for example, on measurable $\mathbb {Z}$–periodic functions on $\mathbb {R}$. Then in all $L^p$ spaces, the Brjuno function is the solution of the linear equation

$$\begin{aligned} {}[(1-T)B_1](x) = -\log x\,. \end{aligned}$$

(1.3)

By exploiting the fact that the operator T as in (1.2) acting on $L^p$ spaces has spectral radius strictly smaller than 1, one can indeed obtain (1.1) by a Neumann series for $(1-T)^{-1}$, see [25] for details.

Local properties of the Brjuno function have been recently investigated by Balazard and Martin [5] and its multifractal spectrum was determined by Jaffard and Martin in [18]. Our results open the path to a possible investigation of the multifractal properties of the k-Brjuno and the Wilton functions as in [18]. The investigation of the local regularity properties of the Brjuno and Wilton functions made by [18] should also be possible for the k-Brjuno and the Wilton functions.

1.2 k-Brjuno functions

We might generalise the Brjuno condition by looking at irrational numbers x fulfilling

$$\begin{aligned} \sum _{n=0}^{\infty } \frac{\log (q_{n+1}(x))}{q_n^k(x)} < \infty , \end{aligned}$$

(1.4)

for some $k\ge 0$. We will call this condition k-Brjuno condition; it has found applications in a range of different settings of which we will give an incomplete list in the following.

If for $k\ge 2$ even, we let $E_k$ be the Eisenstein series of weight k defined in the upper half-plane $\mathbb {H}$, then its Fourier expansion is given by

$$\begin{aligned} E_k(z)=1-\frac{2k}{b_k}\sum _{n=1}^{\infty }\sigma _{k-1}(n)e^{2\pi i nz}, \end{aligned}$$

where $b_k$ is the kth Bernoulli number and $\sigma _{k-1}(n)=\sum _{d|n} d^{k-1}.$ For all $k\ge 4$, $E_k$ is modular of weight k under the action of ${{\,\textrm{SL}\,}}_2(\mathbb {Z})$, and $E_2$ is quasi-modular of weight 2 under the action of ${{\,\textrm{SL}\,}}_2(\mathbb {Z})$, see for example [40]. The function $E_2$ can be viewed as a modular (or Eichler) integral on ${{\,\textrm{SL}\,}}_2(\mathbb {Z})$ of weight 2 with the rational period function $-\frac{2\pi i}{z}$, see for example [19].

For $k \ge 2$ even and $z \in \mathbb {H}$, denote $\varphi _k(z) = \sum _{n=1}^\infty \frac{\sigma _{k-1}(n)}{n^{k+1}}e^{2\pi i nz}.$ We have that

$$\begin{aligned} \sum _{n=1}^{\infty }\sigma _{k-1}(n)e^{2\pi i nz} = \left( \frac{1}{2\pi i}\frac{\partial }{\partial z}\right) ^{k+1}\varphi _k(z), \end{aligned}$$

and

$$\begin{aligned} \varphi _k(z) = \frac{B_k(2\pi i)^{k+1}}{k!2k} \int _{i\infty }^z(z-t)^k(E_k(t)-1)dt. \end{aligned}$$

Consider the imaginary part of $\varphi _k$

$$\begin{aligned} F_{k}(x)= \sum _{n=1}^\infty \frac{\sigma _{k-1}(n)}{n^{k+1}} \sin (2\pi n x) \quad \text {for } x\in \mathbb {R}. \end{aligned}$$

Analytic properties, differentiability and the Hölder regularity exponent, of the function $F_k$ (and the real part of $\varphi _k$) were studied by the third author. It has been proved that for $F_k$ the differentiability is related to a condition resembling the Brjuno condition. Considering the special case $k=2$, the third author proved that if $\sum _{n=0}^{\infty } \frac{\log (q_{n+1}(x))}{q_n^2(x)} < \infty$ and $\lim _{n \rightarrow \infty } \frac{\log (q_{n+4}(x))}{q_n^2(x)} =0$, then $F_{2}$ is differentiable at $x\in \mathbb {R}\setminus \mathbb {Q}$, whereas if $\sum _{n=0}^{\infty } \frac{\log (q_{n+1}(x))}{q_n^2(x)}$ diverges, then $F_{2}$ is not differentiable at $x\in \mathbb {R}\setminus \mathbb {Q}$. It has been conjectured that for all $k\in \mathbb {N}$ even, $F_k$ is differentiable at $x\in \mathbb {R}\setminus \mathbb {Q}$ if and only if it fulfills the k-Brjuno condition, see [30, 31].

Moreover, Rivoal and Seuret [32] were looking at the sum ${\textsf{F}}_s(x) =\sum _{n=1}^\infty \frac{e^{i\pi n^2 x}}{n^s}$ with $s\in (1/2,1]$ for which it was proven by Hardy and Littlewood that it converges almost surely on $[-1,1]$ but not everywhere. For the case $s=1$ Rivoal and Seuret showed that this sum as well as a generalisation of it converges absolutely if $\sum _{n=0}^{\infty } \frac{\log (q_{n+1}(x))}{q_n^{1/2}(x)} < \infty$, i.e. if the 1/2-Brjuno condition holds.

In addition to appearing in the regularity theory of the boundary behaviour of modular integrals and theta series, the k-Brjuno condition is used in the KAM theory of Gevrey flows. For example the $\alpha$-Brjuno-Rüssmann condition introduced in [3] as well as the s-Brjuno arithmetical condition introduced in [12] for one frequency systems turn out to be equivalent to the $1/\alpha$- or 1/s-Brjuno condition.

The occurrence of a condition of this type motivates the following definition.

For $k\in \mathbb {N}$, let

$$\begin{aligned} B_k(x)=\sum _{n=0}^{\infty }(\beta _{n-1}(x))^k\log \left( \frac{1}{G^n(x)}\right) \end{aligned}$$

(1.5)

be called k-Brjuno function.

We will see in Proposition 2.8 that the convergence of $B_k(x)$ is equivalent to x fulfilling the k-Brjuno condition. It is worth mentioning that a generalisation of the k-Brjuno function was also studied in [32] which also converges if and only if its argument fulfills the k-Brjuno condition.

From this equation, we already get the implicit definition in terms of an analogue of the functional equation of the Brjuno function: if $x\in (0,1)$

$$\begin{aligned} B_k(x)= -\log (x)+x^k\cdot B_k(G(x))\,, \end{aligned}$$

see [25, 27]. The k–Brjuno function converges at an irrational x if and only if (1.4) holds, see also Proposition 2.5 for an even stronger statement about the relation between the k-Brjuno function and (1.4). Obviously, for $k=1$, the function in (1.5) gives the Brjuno function introduced before. We also remark that the Diophantine condition introduced in [32, Thm. 1] which ensures the convergence of ${\textsf{F}}_s(x)$ with $s \in (1/2,1)$ can also be obtained by means of a generalization of (1.5): we will show this in Appendix A.

1.3 $\alpha$-continued fractions

Instead of considering the k-Brjuno function with respect to the Gauss map as in (1.5), it is also possible to use $\alpha$-continued fractions instead. The classical Brjuno function associated to $\alpha$-continued fractions was already investigated in [25]. Let $\alpha \in \left[ \frac{1}{2},1\right]$ and let $A_\alpha :(0,\alpha )\rightarrow [0,\alpha ]$ be the transformation of the $\alpha$-continued fractions given by

$$\begin{aligned} A_{\alpha }(x)=\left| \frac{1}{x}-\left\lfloor \frac{1}{x}-\alpha +1\right\rfloor \right| . \end{aligned}$$

(1.6)

For $\alpha =1$, we obtain the Gauss map associated to the regular continued fraction transformation and for $\alpha =1/2$, we obtain the transformation associated to the nearest integer continued fractions. Nakada [29] was the first to consider all these types of continued fractions as a one-parameter family, however, he considered the ‘unfolded’ version of the $\alpha$-continued fraction which is defined by the Gauss map ${\widetilde{A}}_\alpha (x) = \frac{1}{x}-\left\lfloor \frac{1}{x}-\alpha +1\right\rfloor .$ The version as in (1.6) was considered a little later in [35].

Further, let

$$\begin{aligned} B_{k,\alpha }(x)=\sum _{n=0}^{\infty } \left( \beta ^{(\alpha )}_{n-1}(x)\right) ^k \log \left( \frac{1}{A_\alpha ^n(x)}\right) , \end{aligned}$$

where $\beta ^{(\alpha )}_{j}(x)=\prod _{i=0}^j A_\alpha ^i(x)$ for $j\ge 0$ and $\beta ^{(\alpha )}_{-1}(x)=1$, be the generalisation of (1.5) in the sense that we consider the k-Brjuno function not only for the Gauss transformation but also for other $\alpha$-continued fraction transformations $A_{\alpha }$ with $\alpha \in [1/2, 1]$.

For given $\alpha \in [1/2,1]$, any $x\in (0,1]$ has the $\alpha$-continued fraction expansion given by

$$\begin{aligned} x = \dfrac{1}{a_1+\dfrac{\epsilon _1}{\ddots +\dfrac{\epsilon _{j-1}}{a_j+\ddots }}}, \end{aligned}$$

where $a_j:= a_j^{(\alpha )}(x):=\left\lfloor \frac{1}{A_{\alpha }^{j-1}(x)}-\alpha +1\right\rfloor$ and $\epsilon _j:=\epsilon _j^{(\alpha )}(x)$ is the sign of $\frac{1}{A_{\alpha }^{j-1}(x)} - a_j$.

Related to (1.2) we define an operator $T_{k,\alpha }$ acting on $\mathbb {Z}$–periodic measurable functions f such that $f(-x)=f(x)$ for a.e. $x\in (0,1-\alpha )$ as

$$\begin{aligned} T_{k,\alpha } f(x)=x^k f\left( \frac{1}{x}\right) , \quad x\in (0,\alpha ). \end{aligned}$$

(1.7)

It is the operator $T^{(\alpha )}_\nu$ that was introduced in [25], where $\nu$ corresponds to the exponent k in (1.7). It is understood that the function $T_{k,\alpha } f$ is completed outside the interval $(0,\alpha )$ by imposing on $T_{k,\alpha } f$ the same parity and periodicity conditions as those imposed on f.

Then we have

$$\begin{aligned} {}[(1-T_{k,\alpha })B_{k,\alpha }](x) = -\log x, \quad x\in (0,\alpha ) \end{aligned}$$

(1.8)

which follows by a simple calculation.

1.4 Wilton function

Next, we consider the related concept of the Wilton function which is given by

$$\begin{aligned} W(x)=\sum _{n=0}^{\infty } (-1)^n \beta _{n-1}(x)\log \left( \frac{1}{G^n(x)}\right) , \end{aligned}$$

namely by the alternate sign version of the Brjuno function series (1.1) (see Fig. 1 for its graph).

Fig. 1

Numerical computation of the Brjuno function B (left) and of the Wilton function W (right) when $\alpha =1$. The asymmetric logarithmic singularities at rational points provide an intuitive justification for W not belonging to the BMO space, see Sect. 2

(see Fig. 1 for its graph). It converges if and only if it fulfills the Wilton condition

$$\begin{aligned} \left| \sum _{n=0}^\infty (-1)^n \frac{\log (q_{n+1}(x))}{q_n(x)}\right| < \infty , \end{aligned}$$

see [6, Prop. 7] and Remark 2.7 for an even stronger connection between the Wilton function and the Wilton condition. The points of convergence are called Wilton numbers and appear in the work of Wilton, see [36]. Clearly, all Brjuno numbers are Wilton, but not vice versa (it is not difficult to build counterexamples by using the continued fraction).

The function W satisfies the functional equation for $x\in (0,1)$ being a Wilton number:

$$\begin{aligned} W(x)=\log \left( \frac{1}{x}\right) -xW(G(x)), \end{aligned}$$

which by using the same linear operator T as in (1.2) can be written as

$$\begin{aligned} {}[(1+T)W](x) = -\log x. \end{aligned}$$

(1.9)

We can extend the Wilton function in the same way as we did for the k-Brjuno functions, i.e. replacing the Gauss map with the transformation of the $\alpha$-continued fractions: for $\alpha \in \left[ \frac{1}{2},1\right)$ and for all irrational x we define

$$\begin{aligned} W_\alpha (x)=\sum _{n=0}^{\infty } (-1)^n \beta ^{(\alpha )}_{n-1}(x) \log \left( \frac{1}{A_\alpha ^n(x)}\right) . \end{aligned}$$

We then have

$$\begin{aligned} {}[(1-S_{\alpha })W_\alpha ](x) = -\log x, \quad x\in (0,\alpha ), \end{aligned}$$

where the operator $S_{\alpha }=-T_{1,\alpha }$. Also in this case it is understood that $S_{\alpha }$ acts on $\mathbb {Z}$–periodic measurable functions f such that $f(-x)=f(x)$ for a.e. $x\in (0,1-\alpha )$.

The Wilton function and its primitive have been studied recently by Balazard and Martin in terms of its convergence properties [7] and in the context of the Nyman and Beurling criterion, see [5, 6] and [1]. It would also be possible to define k-Wilton functions generalising the Wilton function in the same way as we generalised the Brjuno function to k-Brjuno functions, i.e. studying $W_k(x)=\sum _{n=0}^{\infty } (-1)^n (\beta _{n-1}(x))^k\log \big (\frac{1}{G^n(x)}\big )$. However, we will not explicitly do the calculations for these kinds of functions.

1.5 BMO properties and complex k-Brjuno and Wilton functions.

In [25] the real regularity properties of the Brjuno function were systematically investigated by exploiting the associated functional equation, culminating with the proof that the Brjuno function belongs to the space $\text {BMO}$ of functions with Bounded Mean Oscillation (we refer to [13, 14] for its definition and properties). Here (see Sect. 2) we extend those results to the k-Brjuno function, which turns out to belong to BMO, and to the Wilton function, which does not. Moreover, we consider their generalisations obtained by replacing the Gauss map with $\alpha$-continued fractions [29], $\alpha \in \left[ \frac{1}{2},1\right]$ and we prove that for all $\alpha$ the associated k-Brjuno functions belong to BMO and that the same holds for the associated Wilton functions when $\alpha$ is restricted to the interval $\left[ \frac{1}{2},\frac{\sqrt{5}-1}{2}\right]$.

By Fefferman’s duality theorem, see for example [33, p. 39], one can add an $L^\infty$ function to a BMO function and obtain that the Hilbert transform of the sum will be an essentially bounded function. One is then led to look for a periodic holomorphic function defined on the upper half-plane whose imaginary part, when looked upon $\mathbb {R}$, is the original function. As in [26] we make this construction for the k-Brjuno functions and the Wilton function. The associated complex k-Brjuno functions $\mathcal {B}_k$ and complex Wilton function $\mathcal {W}$ and their properties are summarized in the following Theorem:

Theorem 1.1

• (1) The complex k-Brjuno function $\mathcal {B}_k\,: \, \mathbb {H}\rightarrow \mathbb {C}$ is given by the series

  $$\begin{aligned} \mathcal {B}_k(z)&= -\frac{1}{\pi }\sum _{p/q\in \mathbb {Q}} \Bigg \{-(q'z-p')^k\left[ \text {Li}_2\left( \frac{p'-q'z}{qz-p}\right) - \text {Li}_2\left( -\frac{q'}{q}\right) \right] \nonumber \\&\qquad \qquad \qquad + (p''-q''z)^k\left[ \text {Li}_2\left( \frac{p''-q''z}{qz-p}\right) -\text {Li}_2\left( -\frac{q''}{q}\right) \right] \nonumber \\&\qquad \qquad + \sum _{n=1}^k \frac{1}{nq^n}\Bigg [-(q'z-p')^{k-n}\, \Bigg (\log (1+\frac{q'}{q})-\sum _{i=1}^{n-1}\frac{1}{i}\left( \frac{1}{(1+q'/q)^i}-1\right) \Bigg ) \nonumber \\&\qquad \qquad \qquad \qquad + (p''-q''z)^{k-n}\,\Bigg (\log (1+\frac{q''}{q})-\sum _{i=1}^{n-1}\frac{1}{i}\left( \frac{1}{(1+q''/q)^i}-1\right) \Bigg )\Bigg ] \Bigg \}, \end{aligned}$$

  (1.10)

  where $[\frac{p'}{q'},\frac{p''}{q''}]$ is the Farey interval such that $\frac{p}{q}=\frac{p'+p''}{q'+q''}$ (with the convention $p'=p-1$, $q'=1$, $p''=1$, $q''=0$ if $q=1$) and $\text {Li}_2\left( z\right)$ is the dilogarithm. The complex Wilton function $\mathcal {W}\,: \, \mathbb {H}\rightarrow \mathbb {C}$ is given by the series

  $$\begin{aligned} \mathcal {W}(z) = -\frac{1}{\pi }\sum _{p/q\in \mathbb {Q}}&\Bigg \{(q'z-p')\left[ \text {Li}_2\left( \frac{p'-q'z}{qz-p}\right) -\text {Li}_2\left( -\frac{q'}{q}\right) \right] \nonumber \\&\quad +(p''-q''z)\left[ \text {Li}_2\left( \frac{p''-q''z}{qz-p}\right) -\text {Li}_2\left( -\frac{q''}{q}\right) \right] \nonumber \\&\quad +\frac{1}{q}\log \left( \frac{(q+q')(q+q'')}{q^2}\right) \Bigg \}\,. \end{aligned}$$

  (1.11)

• (2) The real part of $\mathcal {B}_k$ is bounded on the upper half-plane and its non–tangential limit on $\mathbb {R}$ is continuous at all irrational points and has a decreasing jump of $\frac{\pi }{q^k}$ at each rational number $\frac{p}{q}$.

• (3) As one approaches the boundary the imaginary part of $\mathcal {W}$ behaves as follows:

  • (a) if $x\in \mathbb {R}$ is a Wilton number, then $\text {Im}\mathcal {W}(x+w)$ converges to W(x) as $w\rightarrow 0$ in any domain with a finite order of tangency to the real axis;

  • (b) if $x\in \mathbb {R}$ is Diophantine, then in both cases one can allow domains with infinite order of tangency to the real line.

As the Wilton function is not BMO, it also becomes clear by Fefferman’s theorem that its harmonic conjugate can not be an $L^{\infty }$ function.

1.6 Structure of the paper

The paper is organised as follows. In Sect. 2 we state and prove the $\text {BMO}$-properties of the k-Brjuno and Wilton functions and in Sect. 3 we give statements about the truncated real k-Brjuno functions and the truncated real Wilton function which corresponds to the finite part of the sum at rational numbers.

The subsequent sections deal with a complexification of the system. In Sect. 4 we introduce a complex continued fraction algorithm. In Sect. 5 we extend the operators $T_k:=T_{k,1}$ and $S:= S_{1}$ to the complex plane. The main findings concerning these operators are given in Sect. 6. Particularly, Proposition 6.9 indicates how $T_k$ behaves. The analogous behaviour of S can then immediately be deduced by using the fact that $S=-T_1$.

Finally, with the help of these operators, we define the complex k-Brjuno and complex Wilton functions in Sect. 7.

Those proofs which are very similar to the ones in [26] are given in an appendix.

2 $\text {BMO}$ properties of the real k-Brjuno and real Wilton functions

In this section, we study the bounded mean oscillation ($\text {BMO}$) properties of the real k-Brjuno and Wilton functions - both with respect to different transformations $A_{\alpha }$ with $\alpha \in [1/2, 1]$. Before stating the main results of this section, we will first recall the definition of a $\text {BMO}$ function.

Let $L^1_{\text {loc}}(\mathbb {R})$ be the space of the locally integrable functions on $\mathbb {R}$. Recall that the mean value of a function $f\in L^1_{\text {loc}}(\mathbb {R})$ on an interval I is defined as

$$\begin{aligned} f_I=\frac{1}{|I|}\int _{I}f(x) dx. \end{aligned}$$

For an interval U, we say that a function $f\in \text {BMO}(U)$ if

$$\begin{aligned} \Vert f \Vert _{*,U}:=\sup _{I \subset U}\frac{1}{|I|}\int _{I}|f(x)-f_I|dx<\infty . \end{aligned}$$

(2.1)

For further properties of the BMO space, see for example [25, Appendix] and the monographies [13, 14].

In the following, we will state the main properties of this section which show that the $\text {BMO}$ properties fundamentally differ between the k-Brjuno functions and the Wilton function. We first give the statement for k-Brjuno functions.

Proposition 2.1

For all $k\in \mathbb {R}_{>0}$ and all $\alpha \in [1/2, 1]$, the k-Brjuno function $B_{k,\alpha }$ is a $\text {BMO}$ function.

In contrast, for the Wilton function, we have the following statement:

Theorem 2.2

The Wilton function $W=W_1$ is not a $\text {BMO}$ function.

On the other hand, we define for the following $g:=\frac{\sqrt{5}-1}{2}$ and have:

Theorem 2.3

For all $\alpha \in \left[ \frac{1}{2}, g\right]$, the function $W_\alpha$ is a $\text {BMO}$ function.

Before we start with the proofs of the statements above, we first want to give some remarks about them: Proposition 2.1 is an extension of [25, Thm. 3.2] from the classical Brjuno function to k-Brjuno functions.

As a comparison to Fig. 1 showing $B_{1,1}$ and $W_1$, in Fig. 2, some numerical simulations of $W_{\alpha }$ with different values of $\alpha$ are shown: the numerical evidence supports the conjecture that also for $\alpha \in (g,1)$ the function $W_{\alpha }$ is a $\text {BMO}$ function. Indeed, the graph for $\alpha =e-2$ and $\alpha =0.9$ (and nearby values, not displayed in Fig. 1) suggest that the function $W_\alpha$ exhibits a mix of symmetric logarithmic singularities at rational points and jumps, a singular behaviour which is compatible with belonging to BMO. However, unfortunately, the results from Theorem 2.3 can not immediately be transferred to $\alpha \in (g,1)$, see Remark 2.10 for an explanation of what difficulties occur.

Fig. 2

Numerical computations of $W_\alpha$ for $\alpha =1/2$ (upper left), $\alpha =\frac{\sqrt{5}-1}{2}$ (upper right), $\alpha =e-2$ (lower left) and $\alpha =0.9$ (lower right)

2.1 Proof that real k-Brjuno functions are $\text {BMO}$ functions

The main idea of the proof is to prove the statement for $B_{k,1/2}$, see Proposition 2.4, and to show then that $B_{k,\alpha }$ differs from $B_{k,1}$ only by an $L^{\infty }$ function which follows from Propositions 2.5 and 2.8. As the proof follows from very similar arguments as those in [25, Thm. 3.2], we will only describe shortly the necessary changes in the proofs.

We start by introducing

$$\begin{aligned} X_*{} & {} = \{f\in \text {BMO}(\mathbb {R}): f(x+1)=f(x)\nonumber \\{} & {} \quad \text { for all }x\in \mathbb {R}, ~ f(-x)=f(x) \text { for all }x\in [0,1/2]\} \end{aligned}$$

(2.2)

endowed with a norm which is the sum of the BMO seminorm $\Vert \!\cdot \!\Vert _{*,[0,1/2]}$ as in (2.1) and of the $L^2$ norm on the interval (0, 1/2) (w.r.t. the $A_{1/2}$-invariant probability measure). Then one has:

Proposition 2.4

[25, Thm. 3.3] For all $k \in \mathbb {R}_{>0}$, the operator $T_{k,1/2}$ as in (1.7) is a bounded linear operator from $X_*$ to $X_*$ whose spectral radius is bounded above by $(\sqrt{2}-1)^k$.

To proceed, we prove an analog of [25, Prop. 2.3, eq. (iv)].

Proposition 2.5

For all $k\in \mathbb {R}_{>0}$, there exists a constant $C_{1,k} > 0$ such that for all $\alpha \in [1/2,1]$ and $x \in \mathbb {R} \backslash \mathbb {Q}$, one has

$$\begin{aligned} \left| B_{k,\alpha }(x)-\sum _{j=0}^{\infty }\frac{\log q_{j+1}^{(\alpha )}(x)}{\big (q_j^{(\alpha )}(x)\big )^k} \right| <C_{1,k}. \end{aligned}$$

Before we start with the proof, we recall the following property.

Remark 2.6

Let $k\in \mathbb {R}_{>0}$. As in [25, Remark 1.7], by using $\log q_j^{(\alpha )} \lesssim _k (q_j^{(\alpha )})^{k/2}$ for $k\ge 0$, we can show that for all $\alpha \in [1/2,1]$, there are constants $c_{1,k}$ and $c_{2,k}$ such that

$$\begin{aligned} \sum _{j=0}^\infty \frac{\log q_j^{(\alpha )}}{(q_j^{(\alpha )})^k}\lesssim _ k \sum _{j=0}^\infty \frac{1}{(q_j^{(\alpha )})^{k/2}}\le c_{1,k}\quad \text { and }\quad \sum _{j=0}^\infty \frac{\log 2}{(q_j^{(\alpha )})^k}\le c_{2,k}. \end{aligned}$$

Note that $c_{1,k}$ and $c_{2,k}\rightarrow \infty$ as $k\rightarrow 0$.

Proof of Proposition 2.5

For the following calculations we drop the dependence on $\alpha$ and x. We obtain by analogous calculations as in [25, Prop. 2.3, eq. (iv)] that

$$\begin{aligned}&\bigg |-B_{k,\alpha }(x) +\sum _{j=0}^{\infty }\frac{\log q_{j+1}}{q_j^k} \bigg | =\bigg | \sum _{j=0}^{\infty } \beta _{j-1}^k\log \frac{\beta _j}{\beta _{j-1}} + \sum _{j=0}^{\infty }\frac{\log q_{j+1}}{q_{j}^k}\bigg |\\&=\bigg |\sum _{j=0}^{\infty } \beta _{j-1}^k\log (\beta _jq_{j+1}) -\sum _{j=0}^{\infty }\beta _{j-1}^k \log \beta _{j-1}+ \sum _{j=0}^{\infty }\left( \frac{1}{q_{j}^k}-\beta _{j-1}^k\right) \log q_{j+1}\bigg |\\&\le \sum _{j=0}^{\infty } \left| \beta _{j-1}^k\log (\beta _jq_{j+1})\right| +\sum _{j=0}^{\infty }\left| \beta _{j-1}^k \log \beta _{j-1}\right| +\sum _{j=0}^{\infty }\left| \left( \frac{1}{q_{j}^k} -\beta _{j-1}^k\right) \log q_{j+1}\right| . \end{aligned}$$

We have

$$\begin{aligned} \sum _{j=0}^{\infty } \left| \beta _{j-1}^k\log (\beta _jq_{j+1})\right| \le 2^k\sum _{j=0}^{\infty } \frac{\log 2}{q_j^k} \le 2^k c_{2,k}, \end{aligned}$$

where the last estimate follows as in the proof of [25, Prop. 2.3, eq. (iv)] and $c_{2,k}$ is given in Remark (2.6). Furthermore,

$$\begin{aligned} \sum _{j=0}^{\infty }\left| \beta _{j-1}^k \log \beta _{j-1}\right| \le 2^k\sum _{j=0}^\infty \frac{\log q_j + \log 2}{q_j^k} \le 2^k(c_{1,k}+c_{2,k}) \end{aligned}$$

which also follows as in the proof of [25, Prop. 2.3, eq. (iv)] and $c_{1,k}$ is given in Remark (2.6). Since $\frac{1}{q_{j}^k} = \beta _{j-1} + \epsilon _j \frac{q_{j-1}}{q_j}\beta _j$, we have

$$\begin{aligned} \sum _{j=0}^{\infty }\left| \left( \frac{1}{q_{j}^k}-\beta _{j-1}^k\right) \log q_{j+1}\right| \lesssim _k \sum _{j=0}^{\infty }\max \left\{ \beta _{j-1}^{k-1}, \frac{1}{q_{j}^{k-1}}\right\} \left| \epsilon _{j} \frac{q_{j-1}}{q_j} \beta _j \right| \log q_{j+1}.\nonumber \\ \end{aligned}$$

(2.3)

If $k\ge 1$, then (2.3) is bounded above by

$$\begin{aligned} \sum _{j=0}^\infty \beta _j\log q_{j+1} \le 2\sum _{j=0}^\infty \dfrac{\log q_{j+1}}{q_{j+1}} \le 2 c_{1,k}. \end{aligned}$$

If $0<k<1$, then (2.3) is bounded above by

$$\begin{aligned}&\sum _{j=0}^\infty \max \left\{ \frac{1}{\beta _{j-1}^{1-k}}, q_j^{1-k}\right\} \dfrac{q_{j-1}}{q_j}\beta _j\log q_{j+1} \le \sum _{j=0}^\infty \max \left\{ \frac{\beta _{j}}{\beta _{j-1}^{1-k}},\frac{2q_{j-1}}{q_j^k q_{j+1}}\right\} \log q_{j+1} \\&\le \sum _{j=0}^\infty \max \left\{ \beta _{j}^{k},\frac{2}{q_{j+1}^k}\right\} \log q_{j+1} \le \sum _{j=0}^\infty \frac{2^k}{q_{j+1}^k}\log q_{j+1} \le 2^k c_{1,k}. \end{aligned}$$

Remark 2.7

With analogous methods as above using the absolute values of the sum, we also obtain the following statement: There exists a constant $C > 0$ such that for all $\alpha \in [1/2,1]$ and $x \in \mathbb {R} \backslash \mathbb {Q}$, one has

$$\begin{aligned} \left| W_{\alpha }(x)-\sum _{j=0}^{\infty }(-1)^j\frac{\log q_{j+1}^{(\alpha )}(x)}{q_j^{(\alpha )}(x)} \right| <C. \end{aligned}$$

The next proposition is an equivalent of [25, Prop. 2.4].

Proposition 2.8

For all $k\in \mathbb {R}_{>0}$, there exists a constant $C_{2,k} > 0$ such that for all $\alpha \in [1/2, 1]$ and for all $x \in \mathbb {R} \backslash \mathbb {Q}$ one has

$$\begin{aligned} \left| B_{k,\alpha }(x) - \sum _{j=0}^{\infty }\frac{\log q_{j+1}^{(1)}(x)}{(q_j^{(1)}(x))^k} \right| \le C_{2,k}. \end{aligned}$$

Proof

The proof follows completely analogously to that of [25, Prop, 2.4] with the only difference that we use Proposition 2.5 instead of [25, Prop. 2.3] and each $q_j^{(1)}$ (which is denoted by $Q_j$ in [25]) in the denominator is replaced by $(q_j^{(1)})^k$. $\square$

Proof of Proposition 2.1

By Proposition 2.4, $1-T_{k,\alpha }$ is invertible on $X_*$. A $\mathbb {Z}$–periodic even function equal to $-\log x$ on (0, 1/2] is in $X_*$. Thus, $B_{k,1/2}$ is $\text {BMO}$. Since by Proposition 2.8 for all $\alpha \in [1/2,1]$ the k-Brjuno function $B_{k,\alpha }$ differs from $B_{k,1/2}$ only by an $L^{\infty }$ function, $B_{k,\alpha }$ is a $\text {BMO}$ function as well.

2.2 Proofs of the $\text {BMO}$ properties of the Wilton function

Proof of Theorem 2.2

For brevity, in the following we write $O_I(f)=\frac{1}{|I|}\int _{I}|f(x)-f_I|dx$. Furthermore, as we only consider $\alpha =1$, we also always write W instead of $W_1$. By [24, Prop. A.7], if $I_1$ and $I_2$ are two consecutive intervals, then

$$\begin{aligned} O_{I_1\cup I_2}(f)=\frac{|I_1|}{|I_1|+|I_2|}O_{I_1}(f)+ \frac{|I_2|}{|I_1|+|I_2|}O_{I_2}(f)+\frac{2|I_1||I_2|}{(|I_1|+|I_2|)^2}|f_{I_1}-f_{I_2}|.\nonumber \\ \end{aligned}$$

(2.4)

Let $I_{n}:=\left[ -\frac{1}{n},\frac{1}{n}\right] =\left[ -\frac{1}{n},0\right] \cup \left[ 0,\frac{1}{n}\right] =:I_n^-\cup I_n^+$. By (2.4), $O_{I_n}(f)\ge \frac{1}{2}|f_{I_n^-}-f_{I_n^+}|$ for any $f\in L^1_{\text {loc}}(\mathbb {R})$. By [6, Lem. 2], we have

$$\begin{aligned} \int _{0}^{x}W(t)\textrm{d}t=x\log (1/x)+x+O(x^2), \end{aligned}$$

(2.5)

and

$$\begin{aligned} \int _{1-x}^1W(t)\textrm{d}t=-x\log (1/x)-x+O(x^2\log (2/x)), \end{aligned}$$

(2.6)

thus we clearly have

$$\begin{aligned} W_{I_n^+}=\log (n)+1+O(1/n). \end{aligned}$$

Since W is $\mathbb {Z}$–periodic, we also have

$$\begin{aligned} W_{I_n^-}=-\log (n)-1+O(n^{-1}\log (2n)). \end{aligned}$$

Thus,

$$\begin{aligned} O_{I_n}(W)\ge \log (n)+1+O(n^{-1}\log (2n)), \end{aligned}$$

which completes the proof of the theorem.

The proof that k-Wilton functions are not BMO functions would follow in a similar manner with error terms $O(x^{k+1})$ and $O(x^{k+1}\log (2/x))$ in (2.5) and (2.6) instead.

For the proof of Theorem 2.3, we can not use exactly the same strategy as for the proof of Proposition 2.1. The reason is that, as we have seen in Theorem 2.2, $W_1$ is not a $\text {BMO}$ function. Hence, comparing $W_{\alpha }$ for $\alpha <1$ with $W_1$ can not work. Instead, the underlying idea of the proof is to use that $W_{1/2}$, the Wilton function with respect to the nearest integer continued fraction, is a $\text {BMO}$ function and compare $W_{1/2}$ with $W_{\alpha }$ for $\alpha \in [1/2,g]$, see Proposition 2.9.

Let $X_*$ be defined as in (2.2). Since $S_{1/2}=-T_{1,1/2}$ clearly by Proposition 2.4 we have $W_{1/2}\in X_*$. We will first show that the uniform boundedness of $W_{1/2}-W_{\alpha }$, for $\alpha \in [\frac{1}{2},g]$, in the following proposition, and then we will complete the proof of Theorem 2.3.

Proposition 2.9

For $\alpha \in [\frac{1}{2},g]$, we have $W_{1/2}-W_\alpha \in L^\infty$, where $g=\frac{\sqrt{5}-1}{2}$.

Proof

Let $x\in [0,\frac{1}{2}]$. Since $\alpha \in [\frac{1}{2},g]$, we have $2-\frac{1}{\alpha }\le 1-g\le \frac{1}{2+\alpha }\le 1-\alpha$. Recall that

$$\begin{aligned} A_{1/2}(x) = {\left\{ \begin{array}{ll} \frac{1}{x} - k &{} \text { if }x\in (\frac{1}{k+1/2},\frac{1}{k}], \\ (k+1)-\frac{1}{x} &{} \text { if } x\in (\frac{1}{k+1},\frac{1}{k+1/2}], \end{array}\right. } \quad \text { for }k\ge 2. \end{aligned}$$

Since $\alpha \le \frac{1}{1+\alpha }$, we have

$$\begin{aligned} A_{\alpha }(x) = {\left\{ \begin{array}{ll} \frac{1}{x} - k &{} \text { if }x\in (\frac{1}{k+\alpha },\frac{1}{k}], \\ (k+1)-\frac{1}{x} &{} \text { if } x\in (\frac{1}{k+1},\frac{1}{k+\alpha }], \\ 2-\frac{1}{x} &{} \text { if } x\in (\frac{1}{2},\alpha ], \end{array}\right. } \quad \text { for }k\ge 2. \end{aligned}$$

See Fig. 3 for the graphs of $A_{1/2}$ and $A_\alpha$ for a typical $\alpha \in (1/2,g)$.

Fig. 3

The graphs of $A_{1/2}$ and $A_\alpha$ for $\alpha \in [\frac{1}{2},g]$, where $g=\frac{\sqrt{5}-1}{2}$

Let us denote by $(a_n^{(\alpha )},\epsilon _n^{(\alpha )})$ the nth partial quotient of x given as a $\alpha$-continued fraction, $x_n^{(\alpha )} = A_{\alpha }^n(x)$, where $x_0^{(\alpha )} = x$ and $p_n^{(\alpha )}/q_n^{(\alpha )}$ is its nth principal convergent.

We have $\epsilon _1^{(1/2)}\not =\epsilon _1^{(\alpha )}$ if and only if $x\in (\frac{1}{k+\alpha },\frac{1}{k+1/2})$ for some $k\ge 2$. In this case,

$$\begin{aligned} (a_1^{(1/2)},\epsilon _1^{(1/2)})=(k+1,-1),~ (a_1^{(\alpha )},\epsilon _1^{(\alpha )})=(k,1)~ \text { and } ~x_1^{(1/2)}=1-x_1^{(\alpha )}. \end{aligned}$$

Let

$$\begin{aligned} t_i = \frac{1}{3-\frac{1}{3-\frac{1}{\ddots -\frac{1}{3-\frac{1}{2}}}}}\quad \text { and }\quad \frac{r_i}{s_i} = \frac{1}{3-\frac{1}{3-\frac{1}{\ddots -\frac{1}{3}}}}, \end{aligned}$$

where 3 in the continued fraction expansion appears i times. Note that $r_i=s_{i-1}$ and $2-\frac{1}{1- t_i}=\frac{r_i}{s_i}$. More precisely,

$$\begin{aligned}&\{t_i\}_{i\ge 0} = \left\{ \frac{1}{2}, \frac{2}{5}, \frac{5}{13}, \frac{13}{34}, \frac{34}{89},\cdots \right\} \subset \left( 1-g,\frac{2}{5}\right) \cup \left\{ \frac{1}{2}\right\} ,\\&\left\{ \frac{r_i}{s_i}\right\} _{i\ge 0} = \left\{ 0,\frac{1}{3},\frac{3}{8},\frac{8}{21},\frac{21}{55},\cdots \right\} \subset \{0\}\cup \left[ \frac{1}{3},1-g\right) . \end{aligned}$$

Then, $A_{1/2}(t_i)=t_{i-1}$, $A_\alpha (\frac{r_i}{s_i})=A_{1/2}(\frac{r_i}{s_i})=\frac{r_{i-1}}{s_{i-1}}$ for $i\ge 1$ and $t_i \searrow 1-g$ and $\frac{r_i}{s_i} \nearrow 1-g$.

Now, we suppose that n is the minimal index such that

$$\begin{aligned} x_1^{(1/2)}=x_1^{(\alpha )},\, x_2^{(1/2)}=x_2^{(\alpha )},\, \cdots , x_{n-1}^{(1/2)}=x_{n-1}^{(\alpha )}, \, \text {but} \, x_n^{(1/2)}\not =x_n^{(\alpha )}. \end{aligned}$$

Then,

$$\begin{aligned}&(a_{i}^{(1/2)},\epsilon _i^{(1/2)})=(a_i^{(\alpha )},\epsilon _i^{(\alpha )}) \text { and } q_i^{(1/2)} = q_i^{(\alpha )} \,\text {for}\,1\le i\le n-1,\\&x_n^{(1/2)} = 1- x_n^{(\alpha )}, \,a_n^{(1/2)} = a_n^{(\alpha )} + 1 \ge 3,\, \epsilon _n^{(1/2)} = -1, \, \epsilon _n^{(\alpha )} = 1 \text {and } q_n^{(1/2)}\\&\quad = q_n^{(\alpha )} + q_{n-1}^{(1/2)}. \end{aligned}$$

Since $x_n^{(1/2)}\not = x_n^{(\alpha )}$, $x_n^{(1/2)}=1-x_n^{(\alpha )}$ and the domain of $A_{1/2}$ is $[0,\frac{1}{2}]$, we have

$$\begin{aligned} x_n^{(1/2)} \in \left[ 1-\alpha ,\frac{1}{2}\right) \,\text { and } \, x_n^{(\alpha )} \in \left( \frac{1}{2}, \alpha \right] . \end{aligned}$$

(2.7)

Since $[1-\alpha , \frac{1}{2})\subset \bigsqcup _{i=1}^\infty (t_{i},t_{i-1}]$, there is $i\ge 1$ such that $x_n^{(1/2)}\in (t_{i},t_{i-1}]$. This implies

$$\begin{aligned}&x_{n+j}^{(1/2)}\! =\! 3\!-\! \frac{1}{x_{n\!+\!j\!-\!1}^{(1/2)}} \in (t_{i\!-\!j},t_{\!i-\!j\!-\!1}] \text { and } (a_{n+j}^{(1/2)},\epsilon _{n\!+\!j}^{(1/2)}) \!=\! (3,-1) \text { for } 1\le j \le i\!-\!1, \\&x_{n+i}^{(1/2)} = \frac{1}{x_{n+i-1}^{(1/2)}} - 2 \text { and } (a_{n+i}^{(1/2)},\epsilon _{n+i}^{(1/2)}) = (2,1). \end{aligned}$$

On the other hand, $x_n^{(\alpha )}\in [1-t_{i-1},1-t_i)$, which implies

$$\begin{aligned}&x_{n+1}^{(\alpha )} = 2-\frac{1}{x_n^{(\alpha )}}\in \left( \frac{r_{i-1}}{s_{i-1}},\frac{r_{i}}{s_i}\right] \text { and } (a_{n+1}^{(\alpha )},\epsilon _{n+1}^{(\alpha )}) = (2,-1), \\&x_{n+j}^{(\alpha )} \!=\! 3\!-\!S\frac{1}{x_{n+j-1}^{(\alpha )}}\in \left( \frac{r_{i-j}}{s_{i-j}},\frac{r_{i-j+1}}{s_{i-j+1}}\right] \text { and } (a_{n+j}^{(\alpha )},\epsilon _{n+j}^{(\alpha )})\!=\!(3,-1) \text { for } 2\le j \!\le \! i. \end{aligned}$$

Then,

$$\begin{aligned} q_{n+j}^{(1/2)}-q_{n+j}^{(\alpha )} =q_{n+j-1}^{(1/2)},\text { for }1\le j \le i-1,\quad \text {and } q_{n+i}^{(1/2)}=q_{n+i}^{(\alpha )}. \end{aligned}$$

In the following for $g=\left( {\begin{matrix} a &{} b \\ c&{} d \end{matrix}} \right) \in {{\,\textrm{GL}\,}}_2(\mathbb {Z}),$ we denote by $g.z=\frac{az+b}{cz+d}$ the Möbius transform applied on z. With this notation we have

$$\begin{aligned} x_{n+i}^{(1/2)} = \begin{pmatrix} -2&{}1\\ 1&{}0 \end{pmatrix} \begin{pmatrix} 3&{}-1\\ 1&{}0 \end{pmatrix}^{i-1}.x_{n}^{(1/2)}\,\text {and}\, x_{n+i}^{(\alpha )} =\begin{pmatrix} 3&{}-1\\ 1&{}0 \end{pmatrix}^{i-1} \begin{pmatrix} 2&{}-1\\ 1&{}0 \end{pmatrix}.x_{n}^{(\alpha )}. \end{aligned}$$

Since

$$\begin{aligned} \begin{pmatrix} 3&{}-1\\ 1&{}0 \end{pmatrix} \begin{pmatrix} -1&{}1\\ -1&{}2 \end{pmatrix} \begin{pmatrix} 3&{}-1\\ 1&{}0 \end{pmatrix}^{-1} = \begin{pmatrix} -1&{}1\\ -1&{}2 \end{pmatrix}, \end{aligned}$$

$$\begin{aligned} x_{n+i}^{(1/2)}&=\begin{pmatrix} -2&{}1\\ 1&{}0 \end{pmatrix} \begin{pmatrix} 3&{}-1\\ 1&{}0 \end{pmatrix}^{i-1} \begin{pmatrix} -1&{}1\\ 0&{}1 \end{pmatrix}\left( \begin{pmatrix}3&{}-1\\ 1&{}0\end{pmatrix}^{i-1}\begin{pmatrix}2&{}-1\\ 1&{}0\end{pmatrix}\right) ^{-1}.x_{n+i}^{(\alpha )}\\&= \begin{pmatrix}-2&{}1\\ 1&{}0\end{pmatrix}\begin{pmatrix}-1&{}1\\ -1&{}2\end{pmatrix}.x_{n+i}^{(\alpha )} = \begin{pmatrix}1&{}0\\ -1&{}1\end{pmatrix}.x_{n+i}^{(\alpha )}. \end{aligned}$$

It means that $\frac{1}{x_{n+i}^{(1/2)}} = \frac{1}{x_{n+i}^{(\alpha )}}-1$, thus, for $0\le c_1,c_2 \le 1$ and $k\in \mathbb {N}$ such that $k\ge 2$,

$$\begin{aligned} x_{n+i}^{(1/2)}\in \left( \frac{1}{k+c_1},\frac{1}{k+c_2}\right] ~ \text { if and only if }~x_{n+i}^{(\alpha )}\in \left( \frac{1}{k+1+c_1},\frac{1}{k+1+c_2}\right] . \end{aligned}$$

If $x_{n+i}^{(1/2)}\in \left( \frac{1}{k+1},\frac{1}{k+\alpha }\right] \cup \left( \frac{1}{k+1/2},\frac{1}{k+1}\right]$, then

$$\begin{aligned} x_{n+i+1}^{(1/2)} = x_{n+i+1}^{(\alpha )}, ~ a_{n+i+1}^{(1/2)} = a_{n+i+1}^{(\alpha )} - 1,~\epsilon _{n+i+1}^{(1/2)} = \epsilon _{n+i+1}^{(\alpha )}~\text { and } q_{n+i+1}^{(1/2)}=q_{n+i+1}^{(\alpha )}. \end{aligned}$$

If $x_{n+i}^{(1/2)}\in \left( \frac{1}{k+\alpha },\frac{1}{k+1/2}\right]$, then

$$\begin{aligned}&x_{n+i+1}^{(1/2)} = 1- x_{n+i+1}^{(\alpha )}, ~ a_{n+i+1}^{(1/2)} = a_{n+i+1}^{(\alpha )} ,~\epsilon _{n+i+1}^{(1/2)} = -1, ~\epsilon _{n+i+1}^{(\alpha )} = 1\text { and }\\&q_{n+i+1}^{(1/2)}-q_{n+i+1}^{(\alpha )}=q_{n+i}^{(1/2)}. \end{aligned}$$

Thus, by repeating the above process, we conclude that

$$\begin{aligned} q_{j}^{(1/2)}-q_{j}^{(\alpha )} = 0 \text { or } q_{j-1}^{(1/2)} \end{aligned}$$

and if $q_j^{(1/2)}-q_j^{(\alpha )}=q_{j-1}^{(1/2)}$, then $(a_{j+1}^{(1/2)}, \epsilon _{j+1}^{(1/2)})= (3,-1)$ or (2, 1).

Since $a_j^{(1/2)}\ge 2$ and $a_j^{(1/2)}=2$ implies $\epsilon _j^{(1/2)}=1$, we have

$$\begin{aligned} q_j^{(\alpha )} \ge q_j^{(1/2)}-q_{j-1}^{(1/2)} \ge q_{j-1}^{(1/2)}. \end{aligned}$$

(2.8)

Hence,

$$\begin{aligned} \left| \frac{1}{q_j^{(1/2)}}-\frac{1}{q_j^{(\alpha )}}\right| =0~\text { or }~\left| \frac{1}{q_j^{(1/2)}}-\frac{1}{q_j^{(\alpha )}}\right| = \frac{q_{j-1}^{(1/2)}}{q_j^{(1/2)}q_j^{(\alpha )}}\le \frac{1}{q_{j}^{(1/2)}}\le \frac{4}{q_{j+1}^{(1/2)}}.\nonumber \\ \end{aligned}$$

(2.9)

On the other hand, (2.8) also implies

$$\begin{aligned} \left| \log q_{j+1}^{(1/2)}-\log q_{j+1}^{(\alpha )}\right| = \log \left( 1+\frac{q_{j+1}^{(1/2)}-q_{j+1}^{(\alpha )}}{q_{j+1}^{(\alpha )}}\right) \le \log \left( 1+\frac{q_{j}^{(1/2)}}{q_{j+1}^{(\alpha )}}\right) \le \log 2. \end{aligned}$$

Combining this with (2.9) and Remark 2.6, we obtain

$$\begin{aligned}&\left| \sum _{j=0}^\infty (-1)^j\frac{\log q_{j+1}^{(1/2)}}{q_j^{(1/2)}}-\sum _{j=0}^\infty (-1)^j\frac{\log q_{j+1}^{(\alpha )}}{q_{j}^{(\alpha )}}\right| \\&\le \sum _{j=0}^\infty \left| \frac{\log q_{j+1}^{(1/2)}}{q_j^{(1/2)}}-\frac{\log q_{j+1}^{(\alpha )}}{q_{j}^{(1/2)}} \right| +\sum _{j=0}^\infty \left| \frac{\log q_{j+1}^{(\alpha )}}{q_{j}^{(1/2)}} - \frac{\log q_{j+1}^{(\alpha )}}{q_{j}^{(\alpha )}}\right| \\&\le \sum _{j=0}^\infty \frac{\log 2}{q_j^{(1/2)}}+\sum _{j=0}^\infty \frac{4\log q_{j+1}^{(1/2)}}{q_{j+1}^{(1/2)}} \le 4c_1+c_2. \end{aligned}$$

For $x\in (\alpha ,1]$, the values of $W_{1/2}(x)$ and $W_\alpha (x)$ are defined symmetrically as

$$\begin{aligned} W_{1/2}(x) = W_{1/2}(1-x) \text { and } W_\alpha (x)=W_\alpha (1-x). \end{aligned}$$

By Remark 2.7, $W_{1/2}-W_\alpha$ is bounded on $[\alpha ,1]$.

For $x\in (1/2,\alpha ]$, $1-x\in [1-\alpha ,1/2)$. Since $W_{1/2}(x) = W_{1/2}(1-x)$, we can consider that $n=0$ as in (2.7). $\square$

Proof of Theorem 2.3

As we mentioned before, by Proposition 2.4, we have already observed $1-S^{(1/2)}$ is invertible in $X_*$, which together with the fact that the $\mathbb {Z}$–periodic even function equal to $-\log x$ on (0, 1/2] is in $X_*$ implies that $W_{1/2}$ is $\text {BMO}$. By Proposition 2.9, $W_\alpha$ is $\text {BMO}$ for $\alpha \in [1/2,g]$.

Fig. 4

Graph of $A_\alpha$ when $\alpha >g$

Finally, in a remark, we give what difficulties occur if one wants to extend the results of Theorem 2.3 to $\alpha \in (g,1)$.

Remark 2.10

For the case $\alpha >g$, we cannot directly apply the same argument as in the proof of Proposition 2.9. If $\alpha >g$, then $A_\alpha$ has a branch which is defined by $1/x-1$ (see Fig. 4 for the graph of $A_\alpha$) in contrast to the case of $\alpha \le g$. It causes a different behaviour of the orbits of the points under $A_\alpha$. In the proof, we showed a relation between $x_n^{(1/2)}$ and $x_n^{(\alpha )}$. By following the same argument for $\alpha >g$, we only obtain a relation between $x_n^{(1/2)}$ and $x_{n+N}^{(\alpha )}$, where N depends on the number of consecutive points in the orbit of x visiting $[\frac{1}{1+\alpha },\alpha ]$.

3 Behaviour of the truncated real Brjuno function and the truncated real Wilton function

For $x\in \mathbb {R}$, recall that $\beta _{-1}=1$ and

$$\begin{aligned} \beta _j(x)= \{x\}\, G(\{x\})\cdots G^j(\{x\}) = |p_j(\{x\})-q_j(\{x\}) \{x\}|\quad \text { for }j\ge 0, \end{aligned}$$

where $G$ is the Gauss map and $p_j(\{x\})/q_j(\{x\})$ is the jth principal convergent of $\{x\}$ with respect to the regular continued fraction algorithm. Here, unlike in the previous section, we omit the $\alpha$ in $\beta ^{(\alpha )}_j$ as we will always assume $\alpha$ to be one.

In this section, we are interested in comparing a finite k-Brjuno sum or finite Wilton sum to the k-Brjuno or Wilton sum of its principal convergent. To do so we first have to define the finite k-Brjuno or finite Wilton function respectively for a rational number.

Within this section, let $p_j/q_j$ be a rational number whose continued fraction algorithm terminates after r steps, i.e. it can be written as

$$\begin{aligned} p_j/q_j = m_0+\frac{1}{m_1+\frac{1}{m_2+\ddots +\frac{1}{m_r}}} \end{aligned}$$

with $m_r\ge 2$ when $q_{j}>1$. (Of course, this can correspond to the rth principal convergent of a number whose continued fraction expansion starts with $[m_0;m_1,\ldots , m_r,\ldots ]$.) With this, we can define the truncated real Brjuno function by

$$\begin{aligned} B_{k,\textrm{finite}}(p/q)=\sum _{j=0}^{r-1}\left( \beta _{j-1}(p/q-m_0)\right) ^k\log \left( \frac{1}{G^j(p/q-m_0)}\right) \end{aligned}$$

and the truncated real Wilton function by

$$\begin{aligned} W_{\textrm{finite}}(p/q)=\sum _{j=0}^{r-1}(-1)^j\beta _{j-1}(p/q-m_0)\log \left( \frac{1}{G^j(p/q- m_0)}\right) . \end{aligned}$$

Before stating the results of this section, we also introduce the notation $x_j=G^j(x)$ for $x\in (0,1)$. This enables us to state the next lemma which is an analog of [26, Lem. 5.20].

Lemma 3.1

For each $k\in \mathbb {N}$ there exists $C_k>0$ such that for all $x\in (0,1)$ and $r\in \mathbb {N}$, we have

$$\begin{aligned} \left| B_{k,\textrm{finite}}\left( \frac{p_r(x)}{q_r(x)}\right) -\sum _{j=0}^{r-1}\left( \beta _{j-1}(x)\right) ^k\log \frac{1}{x_j}\right| \le C_k x_r (q_r(x))^{-1} \end{aligned}$$

and

$$\begin{aligned} \left| W_{\textrm{finite}}\left( \frac{p_r(x)}{q_r(x)}\right) -\sum _{j=0}^{r-1}(-1)^j\beta _{j-1}(x)\log \frac{1}{x_j}\right| \le C_1 x_r (q_r(x))^{-1}. \end{aligned}$$

Proof

To ease notation we write in the following $p_r$ and $q_r$ instead of $p_r(x)$ and $q_r(x)$ when the dependence on x is clear. If $r=1$, then

$$\begin{aligned}&|B_{k,\textrm{finite}}(p_1/q_1)-\log (1/x)|\\&\quad =|W_{\textrm{finite}}(p_1/q_1)-\log (1/x)| =| \log (q_1/p_1)-\log (1/x)| \\&\quad =| \log (m_1)-\log (m_1+x_1)| = \log (1+ x_1/m_1) \le x_1/m_1. \end{aligned}$$

Suppose that $r>1$, then we have

$$\begin{aligned}&\left| \sum _{j=0}^{r-1}\left[ \left( \beta _{j-1}(p_r/q_r)\right) ^k\log \frac{1}{G^j(p_r/q_r)} - \left( \beta _{j-1}(x)\right) ^k\log \frac{1}{x_j}\right] \right| \nonumber \\&\le \sum _{j=0}^{r-1}\left| \left( \beta _{j-1}(p_r/q_r)\right) ^k\left[ \log \frac{1}{G^j(p_r/q_r)}- \log \frac{1}{x_j}\right] \right| \nonumber \\&\qquad + \sum _{j=0}^{r-1}\left| \left[ \left( \beta _{j-1}(p_r/q_r)\right) ^k- \left( \beta _{j-1}(x)\right) ^k\right] \log \frac{1}{x_j}\right| \end{aligned}$$

(3.1)

and similarly for the Wilton case if $k=1$.

Note that

$$\begin{aligned} \beta _{j-1}(p_r/q_r) = \left| q_{j-1}\frac{p_r}{q_r}-p_{j-1}\right| \end{aligned}$$

(3.2)

for $r\le j-1$. Since $x_j,~G^j(p_r/q_r)\in \left[ \frac{1}{m_{j+1}+1},\frac{1}{m_{j+1}}\right]$, we have $2^{-1} \le \frac{x_j}{G^j(p_r/q_r)}\le 2.$ By using (3.2) and the fact that $\log (y_2/y_1) \le (y_2-y_1)y_1^{-1}$ for $y_1<y_2\in \mathbb {R}_{>0}$, we have

$$\begin{aligned} \left| \log \frac{1}{G^j(p_r/q_r)} - \log \frac{1}{x_j}\right|&\le \max \left\{ G^j(p_r/q_r),x_j\right\} \left| \frac{1}{G^j(p_r/q_r)}-\frac{1}{x_j}\right| \\&\le 2\, G^j(p_r/q_r)\, \frac{|x-p_r/q_r|}{|q_{j}\,\frac{p_r}{q_r}-p_{j}|\,|q_{j}x-p_{j}|}\\&\le \frac{2|x-p_r/q_r|}{|q_{j-1}\,\frac{p_r}{q_r}-p_{j-1}|\,|q_{j}x-p_{j}|}. \end{aligned}$$

Furthermore, we note

$$\begin{aligned} x = \frac{p_{i-1}x_i+p_{i}}{q_{i-1}x_i+q_{i}} \quad \text { and } \quad x_i = - \frac{q_{i}x - p_{i}}{q_{i-1}x-p_{i-1}} \end{aligned}$$

(3.3)

which imply $\prod _{i=0}^\ell (-x_i) = q_\ell x - p_\ell$ and

$$\begin{aligned} |p_\ell -q_\ell x| = \Big |p_\ell - q_\ell \frac{p_{\ell }x_{\ell +1}+p_{\ell +1}}{q_{\ell }x_{\ell +1}+q_{\ell +1}}\Big | = \frac{|p_\ell q_{\ell +1}- q_{\ell }p_{\ell +1}|}{|q_{\ell }x_{\ell +1}+q_{\ell +1}|}< \frac{1}{q_{\ell +1}}. \end{aligned}$$

(3.4)

Hence, for the first summand of (3.1), from (3.2), (3.3) and (3.4), we have

$$\begin{aligned}&\sum _{j=0}^{r-1}\left| \left( q_{j-1}\frac{p_r}{q_r}-p_{j-1}\right) ^k\left[ \log \frac{1}{G^j(p_r/q_r)} - \log \frac{1}{x_j}\right] \right| \\&\le \frac{2}{q_r}\sum _{j=0}^{r-1}\left| q_{j-1}\frac{p_r}{q_r}-p_{j-1}\right| ^{k-1}\left| \frac{p_r-xq_r}{q_{j}x-p_{j}}\right| \le \frac{2x_r}{q_r}\sum _{j=0}^{r-1}x_{j+1}\cdots x_{r-1} \\&\le \frac{2x_r}{q_r}\sum _{j=0}^{r-1} \left( \frac{\sqrt{5}-1}{2}\right) ^{r-j-1} < 2C' x_r q_r^{-1}, \quad \quad \text { (by Proposition 1.4-(iv) in [25])}, \end{aligned}$$

where $C' = \sum _{j=0}^{\infty }\left( \frac{\sqrt{5}-1}{2}\right) ^j=\frac{\sqrt{5}+3}{2}$.

On the other hand, by setting $X_{j,r}=q_{j-1}\,\frac{p_r}{q_r}-p_{j-1}$ and $Y_{j,r}= q_{j-1}x-p_{j-1}$, noting that $|X_{j,r}|\le 1/q_{j-1}$ and $|Y_{j,r}|\le 1/q_{j-1}$ we obtain for the second term of (3.1) that

$$\begin{aligned}&\sum _{j=0}^{r-1}\left| \left( q_{j-1}\,\frac{p_r}{q_r}-p_{j-1}\right) ^k- \left( q_{j-1}x-p_{j-1}\right) ^k\right| \log \frac{1}{x_j} \end{aligned}$$

(3.5)

$$\begin{aligned}&= \sum _{j=0}^{r-1}\left| X_{j,r}-Y_{j,r}\right| \left| X_{j,r}^{k-1}+X_{j,r}^{k-2}Y+\cdots +X_{j,r}Y_{j,r}^{k-2}+Y_{j,r}^{k-1}\right| \log \frac{1}{x_j} \nonumber \\&\le \sum _{j=0}^{r-1} \frac{k}{q_{j-1}^{k-2}}|p_r/q_r-x|\log \frac{1}{x_j} \nonumber \\&\le \sum _{j=0}^{r-1} k q_{j-1}|p_r/q_r-x|\log \frac{1}{x_j}. \end{aligned}$$

(3.6)

Since $\log (y)<y$ and using (3.4) and (3.3), the value in (3.6) is bounded above by

$$\begin{aligned} \frac{k}{q_r} \sum _{j=0}^{r-1} \frac{|p_r-xq_r|}{|q_{j-1}x-p_{j-1}|}\,\frac{1}{x_j} = \frac{k}{q_r} \sum _{j=0}^{r-1} x_{j+1}\cdots x_{r-1}x_{r} \le kC'x_rq_r^{-1}. \end{aligned}$$

By letting $C_k:= 2kC'$, we complete the proof. $\square$

4 Complex continued fractions

We consider a continued fraction on a compact subset of $\mathbb {C}$ which is a complex analog of regular continued fractions. At the beginning of this section we will first define some domains which will be important for defining our complex continued fraction algorithm which we introduce in the sequel.

Similarly as in [26], we consider the following sets:

(4.1)

(4.2)

(4.3)

Fig. 5

(a)$D_0$ and $D_1$. (b)D

(4.4)

(4.5)

(4.6)

(4.7)

Fig. 6

(a)$H_0$, $H_0'$ and $H = H_0\cup H_0'$. (b)$\Delta$

$$\begin{aligned} D_\infty =&\overline{\mathbb {C}}\backslash (D_0\cup \Delta \cup D)\nonumber \\ =&\left\{ z\in \overline{\mathbb {C}} \; | \; |\mathrm{Im(z)}|> \frac{1}{2}\right\} \cup \left\{ z\in \overline{\mathbb {C}} \; | \; \text {Re(z)}> \frac{\sqrt{3}}{2}-1\right\} \nonumber \\&\cup \left\{ z\in \overline{\mathbb {C}} \; | \; \text {Re(z)}>\frac{\sqrt{3}}{2},~\left| z-\frac{\sqrt{3}}{3}\right| >\frac{\sqrt{3}}{3}\right\}&\text {(Figure}{\hbox {7a}}\text {).} \end{aligned}$$

(4.8)

Fig. 7

(a)$D_\infty$. (b)$\iota (D_\infty )$ where $\iota (z)=1/z$

We define an extension of the continued fraction to the complex plane as follows. Let $z\in D$ as in Fig. 5b. Then $1/z \in \bigcup _{n\in \mathbb {N}} (n+\Delta )$, see Figure 6a. If $1/z \in n+\Delta$, then we take $m_1 = n$ and we set $z_1:= 1/z-m_1$. If $z_1 \in \Delta -D$, then we finish the process. If $z_1 \in D$, then we define $m_2$ by an integer n such that $1/z_1 \in n+\Delta$. Repeating this process, we obtain a continued fraction expansion $\{m_i\}_{i=1}^r$ such that

$$\begin{aligned} \frac{1}{z_i}=m_{i+1}+z_{i+1} \quad \text { for all }0< i\le r, \end{aligned}$$

(4.9)

where $z_i\in D$ for $i<r$, and $z_r\in \Delta$. Let $D(m_1,\cdots , m_r)$ be the set of $z_0\in D$ whose first r complex continued fraction entries equal $\{m_i\}_{i=1}^r$ (see Fig. 8).

Note that this continued fraction algorithm does not coincide with Hurwitz’ continued fractions, which are often used on the complex plane, but which are expanded with Gaussian integers [17], see also [16]. The goal here is very different: our complex continued fraction expansion follows as closely as possible the standard continued fraction of neighboring real points. For non-real numbers the iteration of our algorithm leads to an increasing sequence of imaginary parts and it stops after finitely many steps, since “it is not possible anymore to associate a neighboring real point”. We will see in the following section why this continued fraction definition makes sense.

For $z\in \Delta$ and $i\ge 1$, let

$$\begin{aligned} \varepsilon _i = {\left\{ \begin{array}{ll} 0 &{} \text {if } z\in D(m_1,\cdots , m_i) \text { and } m_i = 1, \\ 1 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(4.10)

We note here that the just defined $\varepsilon _i$ is independent of the definition of $\epsilon _i$ defined for the $\alpha$-continued fractions. However, since in the following sections we only consider the regular continued fraction algorithm with the Gauss map $G$ we do not expect it causes confusion for the reader.

Fig. 8

The sets $D(m_1,\cdots ,m_r)$. The left figure is the partition of D by D(n). The right figure is that of the sets D(1, n) in D(1)

Fig. 9

The partition of D by the sets $H(m_1,\cdots ,m_r)$

We define $p_\ell /q_\ell$ by

$$\begin{aligned} \frac{p_\ell }{q_\ell } = \frac{1}{m_1+\frac{1}{m_2 + \frac{1}{\ddots +\frac{1}{m_\ell }}}} \end{aligned}$$

and $q_0 = p_{-1}=q_{-2}=1$ and $p_0 = p_{-2}=q_{-1}=0$. Then,

$$\begin{aligned} z_0 = \frac{p_{i-1}z_i+p_{i}}{q_{i-1}z_i+q_{i}} \quad \text { and } \quad z_i = - \frac{q_{i}z_0 - p_{i}}{q_{i-1}z_0-p_{i-1}}. \end{aligned}$$

(4.11)

By (4.11), we can easily see that

$$\begin{aligned} \prod _{i=0}^\ell (-z_i) = q_\ell z_0 -p_\ell . \end{aligned}$$

(4.12)

We have

$$\begin{aligned} p_\ell -q_\ell z_0= p_\ell - q_\ell \frac{p_{\ell }z_{\ell +1}+p_{\ell +1}}{q_{\ell }z_{\ell +1}+q_{\ell +1}} = \frac{p_\ell q_{\ell +1}- q_{\ell }p_{\ell +1}}{q_{\ell }z_{\ell +1}+q_{\ell +1}} = \frac{(-1)^{\ell +1}}{q_\ell z_{\ell +1}+q_{\ell +1}}.\qquad \end{aligned}$$

(4.13)

Thus, we have

$$\begin{aligned} |p_\ell -q_\ell z_0| < \frac{1}{q_{\ell +1}}. \end{aligned}$$

(4.14)

We set

$$\begin{aligned} H(m_1,\cdots ,m_r) = D(m_1,\cdots ,m_r)\setminus \textrm{int}\left( \bigcup _{m_{r+1}\ge 1}D(m_1,\cdots ,m_{r+1})\right) \qquad \end{aligned}$$

(4.15)

where it is defined by H when $r=0$, see (4.6) for the definition of H. The sets $H(m_1,\cdots ,m_r)$ give a partition of D as in Fig. 9. Then we have

$$\begin{aligned} \big \{z:|\textrm{Im}z|\le 1/2\big \} = \bigcup _{n\in \mathbb {Z}}\bigcup _{r\ge 0}\bigcup _{m_1,\cdots ,m_r\ge 1}\big [H(m_1,\cdots ,m_r)+n\big ]\sqcup \mathbb {R}\backslash \mathbb {Q}, \end{aligned}$$

where the sets in the right-hand term have disjoint interiors. A set $H(m_1,\cdots ,m_r)+n$ meets $\mathbb {R}$ in a unique point which is rational.

5 Complexification of the operators $T_k$ and S

Let J be a closed interval and $k\in \mathbb {N}$. Let $\mathcal {O}^{k}(\overline{\mathbb {C}}\setminus J)$ be the complex vector space of holomorphic functions in $\mathbb {C}\setminus J$, meromorphic in $\overline{\mathbb {C}}\setminus J$ with a zero at infinity of order at least k. There exists $C_{V,k}>0$ such that for each $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus J)$ and each neighbourhood V of J we have

$$\begin{aligned} |\varphi (z)|\le C_{V,k}|z|^{-k}\sup _{\mathbb {C}\setminus \overline{V}}|\varphi | \quad \text {for } z\in C\setminus \overline{V}. \end{aligned}$$

(5.1)

This fact can be easily proven in the following way. We obtain that the function defined as is still analytic on $\mathbb {C}\setminus \overline{V}$. By the maximum principle, we obtain that both functions and $|\varphi |$ attain their maximum on the boundary of $\mathbb {C}\setminus \overline{V}$. Thus, we obtain for all $w\in \mathbb {C}\setminus \overline{V}$

(5.2)

with $C_{V, k}= \sup _{x\in \partial (\mathbb {C}{\setminus } \overline{V})}\left| x\right| ^k$.

Let $\mathcal {O}^{-k}(\overline{\mathbb {C}}\setminus J)$ be the complex vector space of holomorphic functions in $\mathbb {C}\setminus J$, meromorphic in $\overline{\mathbb {C}}\setminus J$ with a pole at infinity of order at most k.

5.1 Hyperfunctions and extensions to spaces of complex analytic functions of the operators $T_k$ and S

Recall that $T_k:=T_{k,\alpha }$ as in (1.7) for given $\alpha \in [1/2,1]$ and $S:=S_{\alpha }$. We want to extend $T_k$ and S for $\alpha =1$ to the space of complex analytic functions.

We proceed as in [26], namely, we extend $T_k$ and S to $A'([0,1])$ the space of hyperfunctions with support contained in [0, 1] for which we will first introduce some definitions. In Proposition 6.1, we will give the definition of these operators and prove that they are indeed well-defined. In Proposition 6.6, we will see that this definition makes sense in terms of $T_k$ and S being a complex extension of $T_k$ and S being studied in the previous sections.

5.1.1 Hyperfunctions

Let K be a non-empty compact set $K\subset \mathbb {R}$. Let us denote by $\mathcal {O}(K)$ the space of functions analytic in a neighbourhood V of K. A hyperfunction with support in K is a linear functional u on $\mathcal {O}(K)$, such that for all neighbourhoods V of K, there is a constant $C_V>0$ such that

$$\begin{aligned} |u(\varphi )|\le C_V \sup _{V}|\varphi |, \quad \forall \varphi \in \mathcal {O}(V). \end{aligned}$$

We denote by $A'(K)$ the space of hyperfunctions with support in K.

Lemma 5.1

[26, Prop. A2.1.] The space $A'([0,1])$ is canonically isomorphic to $\mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$ the complex vector space of holomorphic functions on $\overline{\mathbb {C}}\setminus [0,1]$ vanishing at infinity.

Let us explain the isomorphism in the lemma. Let

$$\begin{aligned} c_z(x)=\frac{1}{\pi (x-z)}. \end{aligned}$$

(5.3)

Given $u\in A'([0,1])$, the corresponding $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$ is obtained by

$$\begin{aligned} \varphi (z)=u(c_z), \quad \forall z\in \mathbb {C}\setminus [0,1]. \end{aligned}$$

(5.4)

On the other hand, for every $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$, the corresponding $u\in A'([0,1])$ is given by

$$\begin{aligned} u(\psi )= \frac{i}{2} \int _{\gamma } \varphi (z)\psi (z)\textrm{d}z, \quad \forall \psi \in \mathcal {O}(V), \end{aligned}$$

(5.5)

where V is a complex neighbourhood of [0, 1] and $\gamma$ is any piecewise $\mathcal {C}^1$ path winding around [0, 1] in the positive direction. That u as in (5.5) and $c_z$ as in (5.3) are inverse to each other can be easily seen by using the substitution $\xi =1/\omega$ resulting in $\textrm{d}\omega =-\xi ^{-2}\textrm{d}\xi$ which gives

$$\begin{aligned} u(c_z)&=\frac{i}{2} \int _{\gamma } \varphi (\omega )c_z(\omega )\textrm{d}\omega =\frac{i}{2} \int _{\gamma } \varphi (\omega )\,\frac{1}{\pi }\,\frac{1}{\omega -z}\,\textrm{d}\omega \\&=\frac{i}{2\pi } \int _{1/\gamma } \frac{\varphi (1/\xi )}{\left( 1/\xi -z\right) (-\xi ^2)}\,\textrm{d}\xi \\&=\frac{1}{z}\int _{1/\gamma }\frac{i}{2\pi }\,\frac{\varphi (\xi ^{-1})\xi ^{-1}}{\xi -1/z}\,\textrm{d}\xi =\frac{1}{z}\, \varphi (z)\,\left( \frac{1}{z}\right) ^{-1} =\varphi (z), \end{aligned}$$

where we denote by $1/\gamma$ the transformed path and notice that $1/\gamma$ goes in the negative direction. We remark that in (5.5) we correct the formula for u given in [26, Appendix 2] where it is given with an additional factor $\pi$. We write $u(x)=\frac{1}{2i}(\varphi (x+i0)-\varphi (x-i0))$ (see [26, Appendix 2] for details).

5.1.2 Formulas of the extensions of $T_k$ and S

We obviously have $S=-T_1$. Hence, for the following, we will introduce the main formulas for $T_k$ only and give explanations where differences occur for S. We also want to remark that if one wanted to study k-Wilton functions their appropriate operator would be $S_k=-T_k$ and most of the following calculations would follow analogously. For $k \in \mathbb {N}$, $m\in \mathbb {N}^*$ and $f\in L^2([0,1])$, let us consider

$$\begin{aligned} T_{k,m}f(x) = \left\{ \begin{array}{ll} x^kf\left( \frac{1}{x}-m\right) , &{} \text {if }x \in \left[ \frac{1}{m+1},\frac{1}{m}\right] ,\\ 0, &{} \text {otherwise}. \end{array} \right. \end{aligned}$$

(5.6)

We then have $T_k=\sum _{m=1}^{\infty }T_{k,m}$. For $\varphi ,~ \psi \in L^2([0,1])$, we define the $L^2$-adjoint $T^*_{k,m}$ by

$$\begin{aligned} \int _{0}^{1}T_{k,m}\varphi (x)\psi (x)dx=\int _{0}^{1}\varphi (x)T^*_{k,m}\psi (x)dx. \end{aligned}$$

(5.7)

Therefore, we have

$$\begin{aligned} T^*_{k,m}\psi (x)=\frac{1}{(m+x)^{k+2}}\psi \left( \frac{1}{m+x}\right) . \end{aligned}$$

(5.8)

By a slight generalisation of [26, (1.10)], we have that if $\psi$ is holomorphic in a neighbourhood V of [0, 1], then $T^*_{k,m}\psi$ is holomorphic in V and we have

$$\begin{aligned} \sup _{V}|T^*_{k,m}\psi |\le \frac{1}{2m^{k+2}}\sup _{V}|\psi |, \end{aligned}$$

where we may take $V= D_\infty ^c$ equipped with the Poincaré metric on the hyperbolic Riemann surface $\overline{\mathbb {C}}{\setminus } [0,1]$. It follows that the series $\sum _{m=1}^{\infty }T_{k,m}u$ converges to a hyperfunction $T_{k}u$ in $A'([0,1])$.

We have

$$\begin{aligned} T_{k,m}^*c_z(x) = - z^k \left( c_{\frac{1}{z}-m}(x) - c_{-m}(x)\right) + \sum _{n=1}^k \frac{z^{k-n}}{n!} \cdot \frac{\partial ^n}{\partial z^n}c_z(x)\Big |_{z=-m}, \end{aligned}$$

where $c_z$ is as in (5.3). For $\varphi \in \mathcal {O}^1(\overline{\mathbb {C}}\backslash [0,1])$, let u be the corresponding hyperfunction in $A'([0,1])$ as in (5.4). Then, $T_{k,m}\varphi$ is defined by $(T_{k,m} u)(c_z)$. If $z\not \in [\frac{1}{m+1},\frac{1}{m}]$, then we have

$$\begin{aligned} (T_{k,m}u)(c_z) = u(T_{k,m}^*c_z). \end{aligned}$$

This follows from (5.7) and the fact that for $z\notin [\frac{1}{m+1},\frac{1}{m}]$ we have that $T_{k,m}u$ and $T_{k,m}^*c_z$ are analytic. Hence, (5.7) already implies the equality.

By (5.4) and (5.5), we have

$$\begin{aligned} u(T_{k,m}^*c_z)&= - z^k \left( u(c_{\frac{1}{z}-m}(x)) - u\left( c_{-m}(x)\right) \right) + \sum _{n=1}^k \frac{z^{k-n}}{n!} u\left( \frac{\partial ^n}{\partial z^n}c_z(x)|_{z=-m}\right) \\&= - z^k\left( \varphi \left( \frac{1}{z} - m\right) -\varphi (-m)\right) + \sum _{n=1}^{k}\frac{z^{k-n}}{n!}\varphi ^{(n)}(-m). \end{aligned}$$

For $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$, the formula for $T_k$ is given by

$$\begin{aligned} T_k\varphi (z) = - \sum _{m=1}^{\infty } z^k\left( \varphi \left( \frac{1}{z}-m\right) -\varphi (-m)\right) + \sum _{m=1}^{\infty }\sum _{n=1}^{k}\frac{z^{k-n}}{n!}\varphi ^{(n)}(-m),\nonumber \\ \end{aligned}$$

(5.9)

which will be shown in Proposition 6.1. By using $S = -T_1$, we can also deduce that a natural extension of S to the space of complex analytic functions is

$$\begin{aligned} S\varphi (z) = \sum _{m=1}^\infty z\left( \varphi \left( \frac{1}{z}-m\right) -\varphi (-m)\right) - \sum _{m=1}^\infty \varphi '(-m). \end{aligned}$$

(5.10)

5.1.3 Algebraic properties of the inversion of $(1-T_k)$ and $(1-S)$, the monoid $\mathcal {M}$ and its actions.

Let us consider the monoid

$$\begin{aligned} \mathcal {M}=\left\{ g=\begin{pmatrix} a &{} b \\ c&{} d \end{pmatrix} \in {{\,\textrm{GL}\,}}_2(\mathbb {Z}) \; | \; d\ge b \ge a \ge 0 \text { and } d\ge c\ge a\ge 0 \right\} \cup \begin{pmatrix}1&{} 0\\ 0 &{} 1\end{pmatrix}, \end{aligned}$$

which is generated by $g(m):= \left( {\begin{matrix}0&{}1\\ 1&{}m\end{matrix}}\right)$ for $m\in \mathbb {N}$. For $g\in \mathcal {M}$ and $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$, we define

$$\begin{aligned} L_{k,g} \varphi (z):= & {} \det (g)^{k+1}\Bigg [(a-cz)^k\varphi \left( \frac{dz-b}{a-cz}\right) \nonumber \\{} & {} \quad -\sum _{n=0}^{k}(a-cz)^{k-n}\frac{\det (g)^{n}}{c^{n} n!}\varphi ^{(n)}\left( -\frac{d}{c}\right) \Bigg ] \end{aligned}$$

(5.11)

and

$$\begin{aligned} \overline{L}_{g} \varphi (z):= & {} \det (g)(a-cz)\left( \varphi \left( \frac{dz-b}{a-cz}\right) -\varphi \left( -\frac{d}{c}\right) \right) \nonumber \\{} & {} \quad -\frac{1}{c}\varphi '\left( -\frac{d}{c}\right) = \det (g) L_{1,g}\varphi (z). \end{aligned}$$

(5.12)

As we will see in Proposition 6.6, we have

$$\begin{aligned} (1-T_k)^{-1}\varphi (z)=\sum _{j=0}^{\infty }T_k^j\varphi (z)=\sum _{g\in \mathcal {M}}L_{k,g}\varphi (z), \end{aligned}$$

(5.13)

and from analogous arguments, we will see that

$$\begin{aligned} (1-S)^{-1}\varphi (z)=\sum _{j=0}^{\infty }S^j\varphi (z)=\sum _{g\in \mathcal {M}}\overline{L}_{g}\varphi (z). \end{aligned}$$

(5.14)

For $g =\big ({\begin{matrix}a&{}b\\ c&{}d \end{matrix}}\big )$ and $n\in \mathbb {Z}$, let

$$\begin{aligned} L_{g}^{(n)}\varphi (z):=\frac{\det (g)^{n+1}}{(a-cz)^n}\varphi \left( \frac{dz-b}{a-cz}\right) \end{aligned}$$

(5.15)

and

$$\begin{aligned} \overline{L}_{g}^{({n})}\varphi (z):=\frac{\det (g)^n}{(a-cz)^{n}}\,\varphi \left( \frac{dz-b}{a-cz}\right) =\det (g)\,L_{g}^{(n)}\varphi (z). \end{aligned}$$

(5.16)

Analogously to [26, (2.2)], we are interested in a connection between $L_{k,g}$ and $L_g^{(k+2)}$ and between $\overline{L}_g$ and $\overline{L}_g^{(3)}$ respectively. In the first case, we obtain by induction that for $t\in \mathbb {N}^*$ and $n \in \mathbb {Z}$, we have

$$\begin{aligned} (L_{g}^{(n)}\varphi )^{(t)} =&\det (g)^{n+1} \sum _{\ell =0}^{t-1} (n+\ell )(n+\ell +1)\cdots (n+t-1) \left( {\begin{array}{c}t\\ \ell \end{array}}\right) \\ {}&c^{t-\ell } \det (g)^\ell L_{g}^{(n+t+\ell )}\varphi ^{(\ell )} + \det (g)^{t+n+1} L_g^{(n+2t)}\varphi ^{(t)}, \end{aligned}$$

where $\sum _{\ell =0}^{-1}$ denotes an empty sum. In particular, for all $k\in \mathbb {N}^*$, we have

$$\begin{aligned} (L_{g}^{(-k)}\varphi )^{(k+1)}=L_{g}^{(k+2)}\varphi ^{(k+1)}. \end{aligned}$$

(5.17)

5.1.4 Polynomial corrections to the actions of $\mathcal {M}$

If $\varphi \in \mathcal {O}^{-k}(\overline{\mathbb {C}}\setminus [0,1])$, then it can be uniquely written as

$$\begin{aligned} \varphi (z)=\xi _kz^k+ \xi _{k-1}z^{k-1}+\cdots + \xi _0+p_{k}(\varphi )(z), \end{aligned}$$

(5.18)

with $p_{k}(\varphi )\in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$. Note that if $\varphi \in \mathcal {O}^{-k}(\overline{\mathbb {C}}\setminus [0,1])$, then $\varphi ^{(k+1)}=p_{k}(\varphi )^{(k+1)}$. If $g =\big ({\begin{matrix}a&{}b\\ c&{}d \end{matrix}}\big ) \in {{\,\textrm{GL}\,}}_2(\mathbb {Z})$ and $\psi (z)=\xi _kz^k+\xi _{k-1}z^{k-1}+\cdots + \xi _0$, then

$$\begin{aligned} L_{g}^{(-k)}\psi (z)= & {} \det (g)^{1-k}\big (\xi _k(dz-b)^k\nonumber \\{} & {} \quad + \xi _{k-1}(dz-b)^{k-1}(a-cz)+\cdots + \xi _0(a-cz)^k \big ), \end{aligned}$$

(5.19)

which means that $p_k( L_g^{(-k)}\psi )=0$. Thus, $p_k$ is a projection from $\mathcal {O}^{-k}(\overline{\mathbb {C}}\setminus [0,1])$ to $\mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$. We have the formula

$$\begin{aligned} L_{k,g}\varphi =p_k(L_{g}^{(-k)}\varphi ) \end{aligned}$$

(5.20)

for $g\in \mathcal {M},~ \varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$ which defines an action of $\mathcal {M}$ on $\mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$, where $p_k$ depends on k. To verify this formula we define $\psi (u){:}{=}(L_{g}^{(-k)}\varphi )\big (\frac{a}{c}-\frac{1}{cu}\big )$. Since $\lim _{z\rightarrow \infty }p_k(L_{g}^{(-k)}\varphi )(z)=0$ is equivalent to $\lim _{u\rightarrow 0} \psi (u)=0$, it is enough to look at a projection of $\psi$. We have

$$\begin{aligned} \psi (u)= \det (g)^{1-k} u^{-k}\varphi \left( \frac{u\det (g)-d}{c}\right) \end{aligned}$$

and if we denote by ${\widetilde{p}}_k$ the projection to a function vanishing at 0, then we obtain

$$\begin{aligned} {\widetilde{p}}_k\left( \psi (u)\right)&= \psi (u)- \det (g)^{1-k} u^{-k} \sum _{n=0}^k \frac{\psi ^{(n)}(0)\, u^n}{n!} \nonumber \\&= \psi (u)- \det (g)^{1-k} \sum _{n=0}^k \frac{\det (g)^n\, \varphi ^{(n)}\left( -\frac{d}{c}\right) \, u^{n-k}}{c^n n!}. \end{aligned}$$

(5.21)

Substituting back $u=(a-cz)^{-1}$ gives (5.20). From the definition of $L_{k,g}$ as in (5.11) and (5.19), the following diagram commutes:

By (5.11) and (5.12), the connection between $\overline{L}_g$ and $\overline{L}_g^{(-1)}$ is similar as they only differ by a factor $\det (g)$ from $L_{1,g}$ and $L_g^{(-1)}$.

By the calculations in (5.21), the $(k+1)$th derivative of the difference between $L_{k,g}\varphi$ and $L_g^{(-k)}\varphi$ is zero for $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$. Thus, it follows from (5.17) that

$$\begin{aligned} (L_{k,g}\varphi )^{(k+1)}(\xi )=L_{g}^{(k+2)}\varphi ^{(k+1)}(\xi ) = \frac{\det (g)^{k+1}}{(a-c\xi )^{k+2}} \varphi ^{(k+1)}(g^{-1}.\xi ), \end{aligned}$$

(5.22)

where $\varphi ^{(k+1)}\in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus [0,1])$. By Taylor’s theorem, we have for every $\omega \in \overline{\mathbb {C}}\backslash [0,1]$ that

$$\begin{aligned} L_{k,g}\varphi (z)&= L_{k,g}\varphi (\omega ) + (L_{k,g}\varphi )'(\omega )(z-\omega )+\cdots +\frac{(L_{k,g}\varphi )^{(k-1)}}{(k-1)!}(z-\omega )^{k-1}\\&\qquad +\int _\omega ^z\frac{(L_{k,g}\varphi )^{(k+1)}(\xi )}{k!}(z-\xi )^kd\xi . \end{aligned}$$

When $\omega$ goes to $\infty$, then only the remainder term will be left since $\frac{(L_{k,g}\varphi )^{(\ell )}(\omega )}{\ell !}(z-\omega )^{\ell }\in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$ for all $0\le \ell \le k-1$. From (5.22), the remainder term can be written as

$$\begin{aligned} \frac{\det (g)^{k+1}}{k!}\int _\infty ^z\frac{(z-\xi )^k}{(a-c\xi )^{k+2}}\varphi ^{(k+1)}(g^{-1}.\xi )d\xi . \end{aligned}$$

By a change of variable with $\xi =\xi (t)= g.\left( -\frac{d}{c}+\frac{\det (g)t}{c(a-cz)}\right)$, which means that $\frac{1}{(a-c\xi )^2}d\xi = \frac{1}{c(a-cz)} dt$ and $\xi = -\frac{(a-cz)}{ct}+\frac{a}{c}$, we then deduce that the remainder term equals

$$\begin{aligned} \frac{\det (g)^{k+1}}{k!}\int _0^1\left( \frac{1-t}{c}\right) ^k\frac{\varphi ^{(k+1)}(-\frac{d}{c} +\frac{\det (g)t}{c(a-cz)})}{c(a-cz)}dt. \end{aligned}$$

Then, for $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1]), g\in \mathcal {M}$ and $z\notin \left[ \frac{b}{d}, \frac{a}{c}\right]$, we have

$$\begin{aligned} L_{k,g}\varphi (z) \!=\! \frac{c^{-(k+1)}\det (g)^{k+1}}{k!}(a-cz)^{-1} \int _0^1(1-t)^k\varphi ^{(k+1)}\left( -\frac{d}{c}\!+\!\frac{\det (g)t}{c(a-cz)}\right) dt. \nonumber \\ \end{aligned}$$

(5.23)

We then have for the special case $g=g(m)$ that

$$\begin{aligned} L_{k,g(m)} \varphi (z)&= -z^k\left( \varphi \left( \frac{1}{z}-m\right) -\varphi \left( -m\right) \right) +\sum _{n=1}^{k}\frac{z^{k-n}}{n!}\varphi ^{(n)}\left( -m\right) \end{aligned}$$

(5.24)

$$\begin{aligned}&=\frac{ (-1)^{k}}{z\,k!}\int _{0}^{1}\varphi ^{(k+1)}\left( \frac{t}{z}-m\right) (1-t)^k\textrm{d}t. \end{aligned}$$

(5.25)

Connected to that we note that by (5.22) $L_{g}^{(k+2)}$ can be represented as

$$\begin{aligned} L_g^{(k+2)}\psi (z)= \frac{\det (g)^{k+1}}{(a-cz)^{k+2}}\psi \left( \frac{dz-b}{a-cz}\right) =\big (L_{k,g} \psi ^{(- (k+1))} \big )^{(k+1)}(z) , \end{aligned}$$

(5.26)

for $\psi \in \mathcal {O}^{-k}(\overline{\mathbb {C}}\setminus [0,1])$, where $\psi ^{(-(k+1))}$ denotes the $(k+1)$th primitive (i.e. an integration between $\infty$ and z) of $\psi$. In the special case that $g=g(m)$, we have that $\det \left( g(m)\right) =-1$ and thus

$$\begin{aligned} L_{g(m)}^{(k+2)} \psi (z)&=-z^{-(k+2)}\psi \left( \frac{1}{z}-m\right) . \end{aligned}$$

(5.27)

For $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$ we can immediately conclude from (5.12), (5.24) and (5.25) that

$$\begin{aligned} \overline{L}_{g(m)} \varphi (z){} & {} = z\left( \varphi \left( \frac{1}{z}-m\right) -\varphi \left( -m\right) \right) -\varphi '\left( -m\right) \nonumber \\{} & {} =\frac{1}{z}\int _{0}^{1}\varphi ''\left( \frac{t}{z}-m\right) (1-t)dt. \nonumber \\ \end{aligned}$$

(5.28)

we restricted ourselves to $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$. However, the same calculations of this section hold true if we consider instead $\gamma _1> \gamma _0>-1$, $I=[\gamma _0,\gamma _1]$, $z\notin \left[ 0,\frac{1}{\gamma _0+m}\right]$, and $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus I)$.

6 Convergence of the sum over the monoid and boundary behaviour

In this Section we will show how to adapt the results of Sections 3 and 4 of [26] to our monoid action. Also in our case the complex continued fraction introduced in Sect. 4 will play a fundamental role.

6.1 Convergence of the sum over the monoid

From the following proposition, we will be able to deduce that $T_k$ is given by (5.9). It is an analog of [26, Prop. 3.1].

Let $-1<\gamma _0<\gamma _1$, $I=[\gamma _0,\gamma _1]$, $J=\Big [0,\frac{1}{(1+\gamma _0)}\Big ]$, and

$$\begin{aligned} U_\varepsilon := \{z\in \mathbb {C}\;|\;\text {re(z)}\in (-\infty ,\gamma _0-\varepsilon ]\cup [ \gamma _1+\varepsilon ,+\infty ) \text { or } \mathrm{Im(z)}\notin (-\varepsilon ,\varepsilon )\}. \end{aligned}$$

(6.1)

Proposition 6.1

We have the following statements.

• (i) For all $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus I)$, the series $\sum _{m=1}^\infty L_{k,g(m)}\varphi$ converges uniformly on compact subsets $K \subseteq \overline{\mathbb {C}}\setminus J$ to a function $T_k\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus J)$ and there exist $\varepsilon >0$ and $C_{K,k}>0$ such that $\displaystyle {\sup _{K}|T_k\varphi |\le C_{K,k}\sup _{U_{\varepsilon }}|\varphi |}.$

• (ii) For all $\psi \in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus I)$, the series $\sum _{m=1}^\infty L_{g(m)}^{(k+2)}\psi$ converges uniformly on compact subsets $K \subseteq \overline{\mathbb {C}}\setminus J$ to a function in $\mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus J)$, which we denote by $T_{k}^{(k+2)}\psi$ and there exist $\varepsilon >0$ and $\overline{C}_{K,k}>0$ such that $\displaystyle {\sup _{K}|T_{k}^{(k+2)}\psi |\le \overline{C}_{K,k} \sup _{U_{\varepsilon }}|\psi |}.$

• (iii) For all $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus I)$, we have $T^{(k+2)}_k\varphi ^{(k+1)}=(T_k\varphi )^{(k+1)}$.

Remark 6.2

Analogous statements can immediately be deduced for $\overline{L}_{g(m)}$ and S by using the relations $\overline{L}_{g(m)}=-L_{1,g(m)}$ and $S=-T$.

Proof of Proposition 6.1

Let $\varepsilon >0$, then, from (5.1), there exists $c_{1,\varepsilon ,k}>0$ such that for all $z\in U_\varepsilon$ and $\psi \in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus I)$ we have $|\psi (z)|\le c_{1,\varepsilon ,k} |z|^{-(k+2)}\sup _{U_\varepsilon }|\psi |$.

If $K \subseteq \overline{\mathbb {C}}\setminus J$ is compact, then there exists $\varepsilon (K)\in \left( 0, \max \{|\gamma _0|, |\gamma _1|\}\right)$ such that $\frac{1}{z}-m\in U_{\varepsilon (K)}$, for all $z\in K$ and $m\in \mathbb {N}$, which implies that $\varepsilon (K)\le |1/z-m|$. Also, there exist $c_{2,K}>0$ and $M_{\gamma _0}\in \mathbb {N}$ such that for all $z\in K$ and $m\ge M_{\gamma _0}$ we have that $\Big |\frac{1}{z}-m\Big |^{-1}\le c_{2,K} m^{-1}$. Therefore, for all $z\in K$ and $\psi \in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus I)$ we have

$$\begin{aligned}&\sum _{m=1}^{\infty }\left| \psi \left( \frac{1}{z}-m\right) \right| \le \sum _{m=1}^{\infty }c_{1, \varepsilon (K)} \left| \frac{1}{z}-m\right| ^{-(k+2)}\sup _{U_{\varepsilon (K)}}|\psi |\nonumber \\ =&c_{1, \varepsilon (K),k}\sup _{U_{\varepsilon (K)}}|\psi |\left( \sum _{m=1}^{M_{\gamma _0}-1} \left( \left| \frac{1}{z}-m\right| ^{-1}\right) ^{k+2}+\sum _{m=M_{\gamma _0}}^{\infty } \left( \left| \frac{1}{z}-m\right| ^{-1}\right) ^{k+2}\right) \nonumber \\ \le&c_{1, \varepsilon (K),k}\sup _{U_{\varepsilon (K)}}|\psi |\left( \sum _{m=1}^{M_{\gamma _0}-1} \left( {\varepsilon (K)}^{-1}\right) ^{k+2}+\sum _{m=M_{\gamma _0}}^{\infty } \left( {c_{2,K}}\, m^{-1}\right) ^{k+2}\right) \nonumber \\ =&c_{1,\varepsilon (K),k}\sup _{U_{\varepsilon (K)}}|\psi |\left( (M_{\gamma _0}-1){\varepsilon (K)}^{-(k+2)} +{c_{2,K}}^{k+2}\sum _{m=M_{\gamma _0}}^{\infty } m^{-(k+2)}\right) \nonumber \\ \le&{\overline{C}_{K,k}}\sup _{U_{\varepsilon (K)}}|\psi |, \end{aligned}$$

(6.2)

with $\overline{C}_{K,k}=c_{2,\varepsilon (K),k}\big ((M_{\gamma _0}-1)\varepsilon (K)^{-(k+2)}+c_{2,K}^{k+2}\sum _{m=M_{\gamma _0}}^{\infty } m^{-(k+2)}\big )$ and we obtain the first part of (i). By integrating $k+1$ times between $\infty$ and z and substituting $\varphi =\psi ^{(-(k+1))}$, we obtain the first part of (ii). The assertion (iii) then follows immediately.

Hence, by (5.22), (5.27) and (6.2) we have for all $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus I)$ and all $z\in K$ that

$$\begin{aligned} |(T_k\varphi )^{(k+1)}(z)|&=|(T_k^{(k+2)}\varphi ^{(k+1)})(z)| \le \sum _{m=1}^{\infty }\left| L_{g(m)}^{(k+2)}\varphi ^{(k+1)}(z)\right| \\&\le |z|^{-(k+2)}\cdot \sum _{k=1}^{\infty }\left| \varphi ^{(k+1)}\left( \frac{1}{z}-m\right) \right| \le \overline{C}_{K,k} |z|^{-(k+2)}\sup _{U_{\varepsilon (K)}}|\varphi ^{(k+1)}|, \end{aligned}$$

implying (ii). We conclude (i) by using Cauchy’s formula.

In particular, we immediately obtain from (5.24) that

$$\begin{aligned} T_k\varphi (z) =- \sum _{m=1}^{\infty } z^k\left( \varphi \left( \frac{1}{z}-m\right) -\varphi (-m)\right) + \sum _{m=1}^{\infty }\sum _{n=1}^{k}\frac{z^{k-n}}{n!}\varphi ^{(n)}(-m) \end{aligned}$$

proving (5.9).

Let $D_{\infty }$ be given as in (4.8) and for $\rho >0$ let

$$\begin{aligned} V_\rho (D_\infty )=\{z\in \overline{\mathbb {C}}\setminus [0,1]\;|\;d_{\text {hyper}}(z,D_\infty )<\rho \}, \end{aligned}$$

where $d_{\text {hyper}}$ denotes the Poincaré metric on $\overline{\mathbb {C}}\setminus [0,1]$.

To define the complex k-Brjuno functions, we deal with $(1-T_k)^{-1}$ as in the real case, see (1.8) and also (7.9). The following proposition is an analogous property to [25, Thm. 2.6] for the real Brjuno function and it is a generalisation of [26, Prop. 3.3] for the case of $k=1$, which guarantees that $(1-T_k)^{-1} = \sum _{r=1}^\infty T_k^r$ converges.

However, before we state this proposition we first give an estimate of the derivatives of $\varphi$ using Cauchy’s integral formula. For $\varphi \in \mathcal {O}^{n}(\overline{\mathbb {C}}\setminus [0,1])$, let . Then we have

By Cauchy’s integral formula, for a circle $\{\omega :|\omega -z|=R\}$ contained in $V_\rho (D_{\infty })$ whose center is given by $z\in V_\rho (D_\infty )$, we have

Since we have

we obtain

$$\begin{aligned} |\varphi ^{(j)}(z)| \le c_{n,j}'\frac{j!\sup _{V_\rho (D_\infty )}|\varphi |}{R^{n+j}},\quad \text {where } c_{n,j}':= \sup _{z\in \partial D_\infty }|z|^{n} \cdot \sum _{i=0}^j {n+i-1 \atopwithdelims ()i}. \nonumber \\ \end{aligned}$$

(6.3)

Proposition 6.3

Let $\rho \ge 0$. The following statements hold.

• (i) For all $k\in \mathbb {N}$, there exists $C_{\rho , k}>0$ such that, for all $r\ge 0$ and $\psi \in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus [0,1])$, we have

  $$\begin{aligned} \sup _{z\in V_\rho (D_\infty )}|((T_k^{(k+2)})^r\psi )(z)|\le C_{\rho ,k}\left( \frac{\sqrt{5}-1}{2}\right) ^{rk} \sup _{z\in V_\rho (D_\infty )}|\psi (z)|. \end{aligned}$$

• (ii) For all $k\in \mathbb {N}$, there exists $\overline{C}_{\rho , k}$ such that, for all $r\ge 0$ and $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$, we have

  $$\begin{aligned} \sup _{z\in V_\rho (D_\infty )}|(T_k^r\varphi )(z)|\le \overline{C}_{\rho , k}\left( \frac{\sqrt{5}-1}{2}\right) ^{rk} \sup _{z\in V_\rho (D_\infty )}|\varphi (z)|. \end{aligned}$$

Similarly to the last proposition, we remark also here that analogous statements can immediately be deduced for $\overline{L}_{g(m)}$ and S by using the relations $\overline{L}_{g(m)}=-L_{1,g(m)}$ and $S=-T$.

The proof of the above proposition is given in the Appendix.

In order to state the next proposition, we define

$$\begin{aligned} Z=\left\{ \begin{pmatrix} 1 &{} n \\ 0&{} 1 \end{pmatrix} \in {{\,\textrm{GL}\,}}_2(\mathbb {Z}) \; | \; n\in \mathbb {Z}\right\} . \end{aligned}$$

(6.4)

From now on, we consider $L_{k,g}$ and $\overline{L}_g$ for $g\in {{\,\textrm{GL}\,}}_2(\mathbb {Z})$. For $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$, since $\varphi _0^{(n)}\in \mathcal {O}^{n+1}(\overline{\mathbb {C}}\setminus [0,1])$, we have $\lim _{c\rightarrow 0}\varphi _0^{(n)}(-d/c)/c^n=0$ for $n\ge 0$. Thus, we define $L_{k,g}$ when $c=0$ by $L_{k,g}\varphi (z) = \det (g)^{k+1}a^k\varphi \left( \frac{dz-b}{a}\right)$. Especially, for $g\in Z$, we have

$$\begin{aligned} L_{k,\left( {\begin{matrix}1&{}n\\ 0&{}1\end{matrix}}\right) }\varphi (z) =\overline{L}_{\left( {\begin{matrix}1&{}n\\ 0&{}1\end{matrix}}\right) }\varphi (z) = \varphi (z-n). \end{aligned}$$

For $H\subset \textrm{GL}_2(\mathbb {Z})$, let us denote

$$\begin{aligned} \sum \nolimits _{H}^{(k+2)}\psi :=\sum _{g\in H}L_{g}^{(k+2)}\psi \quad \text {and}\quad \sum \nolimits _{H,k}\varphi :=\sum _{g\in H}L_{k,g}\varphi , \end{aligned}$$

which are uniformly summable on compact subsets of $\overline{\mathbb {C}}\setminus [0,1]$.

The following three propositions which give a generalisation of [26, Coro. 3.6] will show us a relation between $\sum _{\mathcal {M}, k}$ and $T_k$. They state that $\sum _{\mathcal {M}, k}\varphi$ and $\sum _{Z\cdot \mathcal {M}, k}\varphi$ are well-defined on any compact set of $\overline{\mathbb {C}}{\setminus }[0,1]$ for $\varphi \in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus [0,1])$ and $\sum _{\mathcal {M}, k}$ is the inverse of $(1-T_k)$.

Proposition 6.4

Let $\psi \in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus [0,1])$. We have

$$\begin{aligned} \sum \nolimits _{\mathcal {M}}^{(k+2)}\psi =\sum _{r=0}^\infty (T_k^{(k+2)})^r\psi . \end{aligned}$$

(6.5)

Let $K\subset \overline{\mathbb {C}}\setminus [0,1]$ be compact. Then, for all $k\in \mathbb {N}$, there exist $\varepsilon (K)>0$ and $C_{K,k}$ such that

$$\begin{aligned} \sup _{K}\left| \sum \nolimits _{\mathcal {M}}^{(k+2)}\psi \right| \le C_{K,k}\sup _{U_{\varepsilon (K)}}|\psi |, \end{aligned}$$

(6.6)

where $U_{\varepsilon (K)}$ is as in (6.1) with $\gamma _0=0$ and $\gamma _1=1$.

Furthermore, $\sum \nolimits _{Z\cdot \mathcal {M}}^{(k+2)}\psi$ is uniformly summable on all domains of the form $\{|\text {Re}(z)|<A, |\text {Im}(z)|\ge \delta \}$, for some $A,\delta >0$. It is holomorphic in $\mathbb {C}\setminus \mathbb {R}$, $\mathbb {Z}$-periodic, bounded in the neighbourhood of $\pm i\infty$, and

$$\begin{aligned} \sum \nolimits _{Z\cdot \mathcal {M}}^{(k+2)}\psi =\sum \nolimits _Z^{(k+2)}\sum \nolimits _{\mathcal {M}}^{(k+2)}\psi . \end{aligned}$$

(6.7)

Proof

Equation (6.5) follows from (B.2), and equation (6.6) follows directly from Proposition 6.3 taking $\varepsilon$ and $\rho$ such that $K\subset V_\rho (D_\infty )\subset U_{\varepsilon (K)}\subset \overline{\mathbb {C}}{\setminus }[0,1]$.

Furthermore, making use of (5.26), we have for $g' =\left( {\begin{matrix} 1 &{} n \\ 0 &{} 1 \end{matrix}}\right) \in Z$ that $L_{g'}^{(k+2)}\psi (z)=\psi \left( z-n\right)$. Then, we have for all $g'\in Z$ and $g\in \mathcal {M}$ that $L_{g'\cdot g}^{(k+2)}\psi =L_{g'}^{(k+2)}\left( L_{g}^{(k+2)}\psi \right)$ giving (6.7) in the case at least one side is uniformly summable which we will show in the following.

We have by Proposition 6.1 that $(T_k^{(k+2)})^r\psi \in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus [0,1])$ for any $r\ge 0$. Furthermore, by the above consideration, we have that $\sum \nolimits _Z^{(k+2)}\sum \nolimits _{\mathcal {M}}^{(k+2)}\psi (z)=\sum _{n=1}^{\infty }\left( \sum \nolimits _{\mathcal {M}}^{(k+2)}\psi \left( z-n\right) \right)$. Since $\sum _{n=1}^{\infty }f(z-n)\in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus [0,1])$ holds if $f \in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus [0,1])$, it follows that $\sum \nolimits _Z^{(k+2)}\sum \nolimits _{\mathcal {M}}^{(k+2)}\psi \in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus [0,1])$. $\square$

Proposition 6.5

Let $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$. We have

$$\begin{aligned} \sum \nolimits _{\mathcal {M},k}\varphi =\sum _{r=0}^\infty T_k^r\varphi . \end{aligned}$$

(6.8)

Furthermore, $\sum \nolimits _{Z\cdot \mathcal {M},k}\varphi$ is holomorphic in $\mathbb {C}\setminus \mathbb {R}$, $\mathbb {Z}$-periodic, vanishing at $\pm i\infty$ and we have

$$\begin{aligned} \left( \sum _{\mathcal {M}, k}\varphi \right) ^{(k+1)} =\sum \nolimits _{\mathcal {M}}^{(k+2)}\varphi ^{(k+1)} \quad \text { and } \quad \left( \sum _{Z\cdot \mathcal {M},k}\varphi \right) ^{(k+1)} =\sum \nolimits _{Z\cdot \mathcal {M}}^{(k+2)}\varphi ^{(k+1)}. \end{aligned}$$

(6.9)

Proof

By the definition of $\sum _{H,k}$, $\sum _{H}^{(k+2)}$ and (5.22), the equations in (6.9) hold. Equation (6.8) follows from integrating the expression in Proposition 6.4 with $\psi = \varphi ^{(k+1)}$, combining with 6.1-(iii) and the first equation of (6.9). The remaining statements follow immediately from the properties of $\varphi$ and the uniform summability. $\square$

Proposition 6.6

We have

$$\begin{aligned} (1-T_k)\sum \nolimits _{\mathcal {M},k}&= \sum \nolimits _{\mathcal {M},k}(1-T_k)=\text {id},\quad \text {and }\\ (1-T_k^{(k+2)})\sum \nolimits _{\mathcal {M}}^{(k+2)}&= \sum \nolimits _{\mathcal {M}}^{(k+2)} (1-T_k^{(k+2)})=\text {id}. \end{aligned}$$

Proof

It follows from Propositions 6.4 and 6.5. $\square$

For $H\subset \textrm{GL}_2(\mathbb {Z})$, we denote by

$$\begin{aligned} \overline{\sum }_H^{(3)}\psi := \sum _{g\in H}\overline{L}_{g}^{(3)}\psi \quad \text {and} \quad \overline{\sum }_H\varphi :=\sum _{g\in H}\overline{L}_{g}\varphi , \end{aligned}$$

(6.10)

which are uniformly summable on compact subsets of $\mathbb {C}{\setminus } [0,1]$. As an analog of the statements above and to [26, Coro. 3.6] we obtain the same statements as in Propositions 6.4, 6.5 and 6.6 with $k=1$ where we have to replace $T_1$ by S, $\sum$ by $\overline{\sum }$ and $L_{1,g}$ and $L_{1,g}^{(3)}$ by $\overline{L}_{g}$ and $\overline{L}_{g}^{(3)}$. In particular, it shows that (6.10) is summable when $H=\mathcal {M}$ or $H=Z\cdot \mathcal {M}$.

6.2 Boundary behaviour of the sums $\sum _{\mathcal {M},k}\varphi$ and $\overline{\sum }_{\mathcal {M}}\varphi$ over the monoid

We will consider the boundary behaviour of $\sum _{\mathcal {M},k} \varphi$ for $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$. The following proposition is a generalisation of [26, Prop. 4.1] and explains the behaviour of $\sum _{\mathcal {M},k} \varphi$ near 0.

Proposition 6.7

Let $-1< \gamma _0 < \gamma _1$ and $I=[\gamma _0, \gamma _1]$. Also let

$$\begin{aligned} U = \{ z\in \mathbb {C}\;|\; \text {re(z)} \in (-\infty ,\gamma _0-1]\cup [\gamma _1+1,+\infty )\text { or }\text {Im}(z) \notin (-1/2,1/2)\}. \end{aligned}$$

(6.11)

There exists $C_{I,k} > 0$ such that, for all $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\backslash I)$ and all $z \in D_0\cup H_0\cup H_0'$, we have

$$\begin{aligned} \left| T_k\varphi (z) - \sum _{m =1}^\infty \sum _{n=1}^k \frac{ z^{k-n}}{n!} \varphi ^{(n)}(-m) \right| \le C_{I,k} |z|^k\log (1+|z|^{-1})\sup _U|\varphi |. \end{aligned}$$

(6.12)

Moreover, there exists $C_k>0$ such that, for all $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$ and all $z \in D_0\cup H_0\cup H_0'$, we have

$$\begin{aligned} \left| \sum _{\mathcal {M},k}\varphi (z)-\varphi (z) - \sum _{m =1}^\infty \sum _{n=1}^k \frac{ z^{k-n}}{n!} \left( \sum _{\mathcal {M}, k}\varphi \right) ^{(n)}(-m) \right| \le C_k |z|^k\log (1+|z|^{-1})\sup _{D_\infty }|\varphi |.\nonumber \\ \end{aligned}$$

(6.13)

We will give the proof of this proposition in much detail as it differs significantly from the case $k=1$.

Proof

As the proofs of (6.12) and (6.13) are similar, we will first prove (6.13) and later give the differences for the proof of (6.12). For the following, we set ${\widetilde{\varphi }}= \sum _{\mathcal {M}, k}\varphi$. Then, Proposition 6.6 and Proposition 6.1-(i) imply

$$\begin{aligned} {\widetilde{\varphi }}=\varphi +T_k{\widetilde{\varphi }}=\varphi +\sum _{m=1}^{\infty }L_{k,g(m)}{\widetilde{\varphi }}. \end{aligned}$$

By the above equation and (5.9), the expression on the left-hand side of (6.13) can thus be summarized as

$$\begin{aligned}&\left| \sum _{\mathcal {M},k}\varphi (z)-\varphi (z) - \sum _{m =1}^\infty \sum _{n=1}^k \frac{z^{k-n}}{n!} \left( \sum _\mathcal {M}\varphi \right) ^{(n)}(-m) \right| \nonumber \\&=\left| T_k{\widetilde{\varphi }}(z) - \sum _{m =1}^\infty \sum _{n=1}^k \frac{z^{k-n}}{n!} {\widetilde{\varphi }}^{(n)}(-m)\right| =\left| \sum _{m=1}^{\infty }z^k\left( {\widetilde{\varphi }}\left( \frac{1}{z}-m\right) -{\widetilde{\varphi }}(-m)\right) \right| . \end{aligned}$$

(6.14)

We will split the sum in the following way:

$$\begin{aligned}&\left| \sum _{\mathcal {M},k}\varphi (z)-\varphi (z) \sum _{m =1}^\infty \sum _{n=1}^k \frac{z^{k-n}}{n!} \left( \sum _{\mathcal {M},k}\varphi \right) ^{(n)}(-m) \right| \nonumber \\&\le \left| \sum _{m=1}^{\left\lfloor 3|z|^{-1}\right\rfloor +1}z^k\left( {\widetilde{\varphi }}\left( \frac{1}{z}-m\right) -{\widetilde{\varphi }}(-m)\right) \right| \end{aligned}$$

(6.15)

$$\begin{aligned}&\qquad +\left| \sum _{m=\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty } \sum _{n=1}^k \frac{z^{k-n}}{n!} {\widetilde{\varphi }}^{(n)}(-m)\right| \end{aligned}$$

(6.16)

$$\begin{aligned}&\qquad + \left| \sum _{m=\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty }z^k\left( {\widetilde{\varphi }}\left( \frac{1}{z}-m\right) -{\widetilde{\varphi }}(-m)\right) - \sum _{m =\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty } \sum _{n=1}^k \frac{z^{k-n}}{n!} {\widetilde{\varphi }}^{(n)}(-m)\right| . \end{aligned}$$

(6.17)

We first estimate the first summand (6.15). We note that for $m\in \mathbb {N}_{\le \left\lfloor 3|z|^{-1}\right\rfloor +1}$ and $z\in D_0\cup H_0\cup H_0'$ we have that $-m\in D_\infty$ and $1/z-m\in D_{\infty }$.

Applying then (5.1) with $\mathbb {C}\backslash \overline{V}=D_{\infty }$ yields

$$\begin{aligned}&\left| \sum _{m=1}^{\left\lfloor 3|z|^{-1}\right\rfloor +1} z^k\left( {\widetilde{\varphi }}\left( \frac{1}{z}-m\right) -{\widetilde{\varphi }}(-m)\right) \right| \nonumber \\&\le \sum _{m=1}^{\left\lfloor 3|z|^{-1}\right\rfloor +1} c_{1,k} \left| z\right| ^k\cdot \left( \frac{1}{m^k}+\left| -m+\frac{1}{z}\right| ^{-k}\right) \cdot \sup _{D_{\infty }}\left| {\widetilde{\varphi }}\right| \nonumber \\&\le c_{2,k}\left| z\right| ^k\cdot \left( \sum _{m=1}^{\left\lfloor 3|z|^{-1}\right\rfloor +1} \left( \frac{1}{m}+\left| -m+\frac{1}{z}\right| ^{-1}\right) \right) \cdot \sup _{D_{\infty }}\left| {\widetilde{\varphi }}\right| \end{aligned}$$

with $c_{2,k}=c_{1,k} \sup _{z\in D_0\cup H_0\cup H_0', m\in \mathbb {N}}\left| -m+\frac{1}{z}\right| ^{-k+1}=c_{1,k}\, 2^{k-1}$. Noting that for all $z\in D_0\cup H_0\cup H_0'$ we have

$$\begin{aligned}&\sum _{m=1}^{\left\lfloor 3|z|^{-1}\right\rfloor +1} \frac{1}{m} \le 1+ \int _1^{\left\lfloor 3|z|^{-1}\right\rfloor +1} \frac{1}{m}\, \textrm{d}m \le 1+ \log ( \left\lfloor 3|z|^{-1}\right\rfloor +1), \end{aligned}$$

and since $\left| \frac{1}{z}-\left( \left\lfloor \text {Re}\left( \frac{1}{z}\right) \right\rfloor -n\right) \right| \ge \left| \frac{1}{z}-\left( \left\lfloor \text {Re}\left( \frac{1}{z}\right) \right\rfloor +n\right) \right|$ for $1\le n \le \lfloor \text {Re}(1/z)\rfloor$ and $\text {Re}(z)\ge 0$, we have

$$\begin{aligned} \sum _{m=1}^{\left\lfloor 3|z|^{-1}\right\rfloor +1} \left| m-\frac{1}{z}\right| ^{-1}&= \sum _{\begin{array}{c} \{ m:1\le m\le \lfloor 3|z|^{-1}\rfloor +1, \\ \left| m- \lfloor \text {Re}(1/z)\rfloor \right| \le 2\} \end{array}} \left| m-\frac{1}{z}\right| ^{-1}\\&\quad + \sum _{\begin{array}{c} \{ m:1\le m\le \lfloor 3|z|^{-1}\rfloor +1, \\ |m- \lfloor \text {Re}(1/z)\rfloor |> 2\} \end{array}} \left| m-\frac{1}{z}\right| ^{-1} \\&\le 10 + 2 \sum _{m=\lfloor \text {Re}(1/z)\rfloor +3}^{\left\lfloor 3|z|^{-1}\right\rfloor +1}\frac{1}{m-\lfloor \text {Re}(1/z)\rfloor -1} \\&\le 10 + 2 \int _{m=1}^{\left\lfloor 3|z|^{-1}\right\rfloor -\lfloor \text {Re}(1/z)\rfloor }\frac{1}{m}\,\textrm{d}m\\&\le 10 + 2 \log (\left\lfloor 3|z|^{-1}\right\rfloor - \lfloor \text {Re}(1/z)\rfloor ). \end{aligned}$$

The last two inequalities even hold true if $\text {Re}(1/z)<0$. The fact that $\lfloor 3|z|^{-1}\rfloor +1 \le (|z|^{-1}+1)^3$ implies that there exists another constant $c_{3,k}>0$ such that, for all ${\widetilde{\varphi }}\in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$ and $z\in D_0\cup H_0\cup H_0'$, we have

$$\begin{aligned} \left| \sum _{m=1}^{\left\lfloor 3|z|^{-1}\right\rfloor +1}z^{k}\left( {\widetilde{\varphi }}\left( \frac{1}{z}-m\right) -{\widetilde{\varphi }}(-m)\right) \right|&\le c_{3,k} \left| z\right| ^k\cdot \log \left( 1+\left| z\right| ^{-1}\right) \cdot \sup _{D_{\infty }}\left| {\widetilde{\varphi }}\right| . \end{aligned}$$

(6.18)

For the second summand (6.16), from (6.3), for all $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$ and $z\in D_0\cup H_0\cup H_0'$ we have

$$\begin{aligned}&\left| \sum _{m=\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty } \sum _{n=1}^k \frac{ z^{k-n}}{n!} {\widetilde{\varphi }}^{(n)}(-m)\right| \\&\quad \le \sum _{m=\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty } \sum _{n=1}^k \frac{|z|^{k-n}}{n!}\cdot c_{k,n}'n!\cdot \frac{1}{m^{k+n}}\cdot \sup _{D_{\infty }}\left| {\widetilde{\varphi }}\right| \nonumber \\&\quad \le \sum _{m=\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty } |z|^{k-1}\cdot 2^{k-1}c_{k,n}' \cdot \frac{1}{m^{k+1}}\cdot \sup _{D_{\infty }}\left| {\widetilde{\varphi }}\right| . \end{aligned}$$

Noting that

$$\begin{aligned} \sum _{m=\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty } m^{-k-1}< \int _{\left\lfloor 3|z|^{-1}\right\rfloor +1}^{\infty }m^{-k-1}\textrm{d}m = {k^{-1}} \left( \left\lfloor 3|z|^{-1}\right\rfloor +1\right) ^{-k} < \left( \frac{|z|}{3}\right) ^{k} \end{aligned}$$

(6.19)

yields that for all $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$ and $z\in D_0\cup H_0\cup H_0'$ we have

$$\begin{aligned} \left| \sum _{m=\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty } \sum _{n=1}^k \frac{z^{k-n}}{n!}\, {\widetilde{\varphi }}^{(n)}(-m)\right|&\le |z|^{2k-1}\cdot c_{k,n}'\cdot \sup _{D_{\infty }}\left| {\widetilde{\varphi }}\right| . \end{aligned}$$

(6.20)

Finally, we estimate the last summand (6.17) using (5.25)

$$\begin{aligned}&\left| \sum _{m=\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty }z^{k}\left( {\widetilde{\varphi }}\left( \frac{1}{z}-m\right) -{\widetilde{\varphi }}(-m)\right) -\sum _{m = \left\lfloor 3|z|^{-1}\right\rfloor +2}^\infty \sum _{n=1}^k \frac{z^{k-n}}{n!} {\widetilde{\varphi }}^{(n)}(-m)\right| \nonumber \\&\le \sum _{m=\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty }\left| \frac{1}{z\, k!}\int _{0}^{1}{\widetilde{\varphi }}^{(k+1)}\left( \frac{t}{z}-m\right) (1-t)^k\textrm{d}t\right| . \end{aligned}$$

(6.21)

Using Cauchy’s integral formula yields for R sufficiently small (possibly depending on m and z) to be defined later that

$$\begin{aligned}&\left| \int _{0}^{1}{\widetilde{\varphi }}^{(k+1)}\left( \frac{t}{z}-m\right) (1-t)^k\textrm{d}t\right| \\&\le \left| \int _0^1 \frac{(k+1)!}{2\pi i}\int _{\left| \omega -\left( \frac{t}{z}-m\right) \right| =R}\frac{{\widetilde{\varphi }}\left( \omega \right) }{\left( \omega -\left( \frac{t}{z}-m\right) \right) ^{k+2}}\textrm{d}\omega \cdot (1-t)^k\textrm{d}t\right| \\&\le \sup _{\left| \omega -\left( \frac{t}{z}-m\right) \right| =R} \big |{\widetilde{\varphi }}\left( \omega \right) \big |\cdot \frac{(k+1)!}{R^{k+2}}\int _0^1(1-t)^k\textrm{d}t =\sup _{\left| \omega -\left( \frac{t}{z}-m\right) \right| =R}\big |{\widetilde{\varphi }}\left( \omega \right) \big |\cdot \frac{2\pi k!}{R^{k+1}}. \end{aligned}$$

Now, we specify the radius R. It is sufficient that $-m+t/|z|+R<\sqrt{3}/2-1$ is fulfilled, i.e. it is sufficient that $R< \sqrt{3}/2-1+m-1/|z|$ is fulfilled and thus we can choose $R= m/3$. Hence, using the above estimate and inserting it into (6.21) yields for all $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$

$$\begin{aligned}&\left| \sum _{m= \left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty }z^k\left( {\widetilde{\varphi }}\left( \frac{1}{z}-m\right) -{\widetilde{\varphi }}(-m)\right) -\sum _{m =\left\lfloor 3|z|^{-1}\right\rfloor +2}^\infty \sum _{n=1}^k \frac{z^{k-n}}{n!} {\widetilde{\varphi }}^{(n)}(-m)\right| \nonumber \\&\le \sum _{m=\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty } 2\pi \cdot 3^{k+1} \cdot |z|^{-1}\cdot m^{-k-1}\cdot \sup _{\left| \omega -\left( \frac{t}{z}-m\right) \right| = m/3} \left| {\widetilde{\varphi }}\right| . \end{aligned}$$

(6.22)

Next, we further estimate $\sup _{\left| \omega -\left( \frac{t}{z}-m\right) \right| =m/3} \left| {\widetilde{\varphi }}\right|$. Using (5.1) and noting that we have $D_{z,m,t}:=\{\omega : \left| \omega -\left( \frac{t}{z}-m\right) \right| = m/3\}\subset D_{\infty }$ and thus $\inf _{\omega \in D_{z,m,t}}|\omega | \ge \left| \frac{t}{z}-m\right| -\frac{m}{3}\ge m- |z|^{-1}-\frac{m}{3}\ge \frac{m}{3}$ we obtain that there exists $c_{4,k}$ such that

$$\begin{aligned} \sup _{D_{z,m,t}} \left| {\widetilde{\varphi }}\right|&\le c_{4,k}\,\sup _{D_{\infty }} \left| {\widetilde{\varphi }}\right| \cdot \sup _{D_{z,m,t}} \left| \omega \right| ^{-k} \le c_{4,k}\,\sup _{D_{\infty }} \left| {\widetilde{\varphi }}\right| \cdot \left( \frac{m}{3}\right) ^{-k} \end{aligned}$$

(6.23)

and for $m\ge \left\lfloor 3|z|^{-1}\right\rfloor +2$, we have

$$\begin{aligned} \sup _{D_{z,m,t}} \left| {\widetilde{\varphi }}\right|&\le c_{4,k} \,\sup _{D_{\infty }} \left| {\widetilde{\varphi }}\right| \cdot \left| z\right| ^k. \end{aligned}$$

Combining this with (6.22) and (6.19) yields

$$\begin{aligned}&\left| \sum _{m=\left\lfloor 3|z|^{-1}\right\rfloor +2}^{\infty }z^k\left( {\widetilde{\varphi }}\left( \frac{1}{z}-m\right) -{\widetilde{\varphi }}(-m)\right) -\sum _{m =1}^\infty \sum _{n=1}^k \frac{z^{k-n}}{n!} {\widetilde{\varphi }}^{(n)}(-m)\right| \nonumber \\&\le 2 \pi c_{4,k}\cdot 3^{k+1} \cdot |z|^{-1}\cdot \left( \frac{|z|}{3}\right) ^{2k} \cdot \sup _{D_{\infty }} \left| {\widetilde{\varphi }}\right| \le 2\pi c_{4,k}\cdot |z|^{2k-1}\cdot \sup _{D_{\infty }}\left| {\widetilde{\varphi }}\right| . \end{aligned}$$

(6.24)

Combining the sum of (6.15), (6.16), and (6.17) estimated in (6.18), (6.20), and (6.24), we obtain that there exists a constant $C_k>0$ such that for all $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$ and $z\in D_0\cup H_0\cup H_0'$ we have the statement of (6.13).

The left-hand side of (6.14) equals the left-hand side of (6.12) if we replace $\varphi$ by ${\widetilde{\varphi }}$. All the remaining calculations stay the same with the difference that we have to take the supremum over U instead of $D_{\infty }$. However, there are only two instances where this is important. The first is the choice of R which has to fulfill $R< \gamma _0+m-1/|z|$ instead of $R< \sqrt{3}/2-1+m-1/|z|$ and for both $R\le m/3$ is sufficient. The second is the occurrence of $\inf _{x\in D_{\infty }} \left| x\right| ^{-k}$ in (6.23) which we estimated by $c_{1,k}$ and which has to be changed to an estimate by a constant depending on I and k. $\square$

Our next concern will be the behaviour of $\sum _{\mathcal {M},k}\varphi$ for $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$ near 1. The following proposition is a generalisation of [26, Prop. 4.2].

Proposition 6.8

There exists $C_k>0$ such that for all $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$ and all $z \in D_1$ we have

$$\begin{aligned} \left| T_k\varphi (z) + z^k \varphi \left( \frac{1}{z}-1\right) - \sum _{m =1}^\infty \sum _{n=1}^k \frac{z^{k-n}}{n!} \varphi ^{(n)}(-m) \right| \le C_k |z-1|\sup _{D_\infty }|\varphi |.\nonumber \\ \end{aligned}$$

(6.25)

Proof

We first notice that (5.9) implies

$$\begin{aligned}&\left| T_k\varphi (z) + z^k\varphi \left( \frac{1}{z}-1\right) - \sum _{m =1}^\infty \sum _{n=1}^k \frac{z^{k-n}}{n!}\varphi ^{(n)}(-m) \right| \nonumber \\&=\left| \sum _{m=2}^{\infty } z^k\varphi \left( \frac{1}{z}-m\right) - \sum _{m=1}^{\infty } z^k \varphi (-m)\right| \nonumber \\&\le |z|^k\cdot \left| \sum _{m=2}^{\infty } \left( \varphi \left( \frac{1}{z}-m\right) -\varphi (1-m)\right) \right| . \end{aligned}$$

(6.26)

For $z \in D_1$, we have $1/z - 1 \in D_0$. Equation (6.3) implies that there exists a constant $c_{1,k}>0$ such that $|\varphi '(w)| \le c_{1,k} m^{-k-1}\sup _{D_{\infty }}|\varphi |$. This implies

$$\begin{aligned}&\left| T_k\varphi (z)+ z^k\varphi \left( \frac{1}{z}-1\right) - \sum _{m =1}^\infty \sum _{n=1}^k \frac{z^{k-n}}{n!}\varphi ^{(n)}(-m) \right| \\&\le |z|^k\cdot \sum _{m=2}^{\infty } c_{1,k} m^{-k-1}\sup _{D_{\infty }}|\varphi |\cdot \left| \frac{1}{z}-1\right| = |z|^{k-1}\cdot \left| 1-z\right| \sup _{D_{\infty }}|\varphi | \sum _{m=2}^{\infty } c_{1,k} m^{-k-1}\\&= C_k \cdot \left| 1-z\right| \sup _{D_{\infty }}|\varphi |, \end{aligned}$$

where $C_k {:}{=}\sup _{z\in D_1}|z|^{k- 1} c_{1,k} \sum _{m=2}^{\infty } m^{-k-1}$. $\square$

We can finally use the previous results and the complex continued fraction algorithm to study the behaviour of $T_k^n\varphi (z)$ when z is close to the boundary. We recall that $D(m_1,\cdots , m_n)$ is the set of $z_0\in D$ whose first n complex continued fraction entries equal $\{m_i\}_{i=1}^n$. Next, we will prove a proposition which is an analog of [26, Prop. 4.4].

Proposition 6.9

For all $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$, $n\in \mathbb {N}$, and $z_0\in D(m_1,\cdots , m_n)$ we have

$$\begin{aligned}{} & {} T_k^n\varphi (z_0)= (-1)^{n(k+1)}(p_{n-1}-q_{n-1}z_0)^{k}(\varphi (z_n)+\varphi (z_n-1)+\varepsilon _n\varphi (z_n+1))\nonumber \\{} & {} - (-1)^{(n-1)(k+1)} (p_{n-2}-q_{n-2}z_0)^{k}(1+z_{n-1})^{k}\varepsilon _{n-1}\varphi \left( -\frac{z_{n-1}}{1+z_{n-1}}\right) +R^{[n]}_{k}\varphi (z_0). \nonumber \\ \end{aligned}$$

(6.27)

The remainder term $R^{[n]}_k$ is holomorphic in the interior of $D(m_1,\cdots , m_n)$ and continuous in $D(m_1,\cdots , m_n)$ and there exists $C_k>0$ such that for all $\varphi \in \mathcal {O}^{k}(\overline{\mathbb {C}}\setminus [0,1])$, $n\in \mathbb {N}$, and $z_0\in D(m_1,\cdots , m_n)$ we have

$$\begin{aligned} \left| R^{[n]}_{k}\varphi (z_0)\right| \le C_k\,n \left( \frac{\sqrt{5}-1}{2}\right) ^{nk}\sup _{D_\infty }|\varphi |. \end{aligned}$$

(6.28)

Proof

We start by an estimate for $n=1$. For $z_0 \in D(m_1)$, by definition, we have

$$\begin{aligned} T_k\varphi (z_0)&= - \sum _{m=1}^{\infty } z_0^k\left( \varphi \left( \frac{1}{z_0}-m\right) -\varphi (-m)\right) + \sum _{m=1}^{\infty }\sum _{j=1}^{k}\frac{z_0^{k-j}}{j!}\varphi ^{(j)}(-m)\\&= -z_0^k(\varphi (z_1)+\varphi (z_1-1)+\varepsilon _1\varphi (z_1+1)) + R_{k}^{(m_1)}(\varphi )(z_0), \end{aligned}$$

with

$$\begin{aligned} R^{(m_1)}_{k}(\varphi )(z_0)&= \sum _{\begin{array}{c} m\ge 1\\ |m-m_1|\le 1 \end{array}}z_0^k \varphi (-m) + \sum _{m=1}^\infty \sum _{j=1}^{k}\frac{z_0^{k-j}}{j!}\varphi ^{( j)}(-m)\\&\qquad - \sum _{\begin{array}{c} m\ge 1\\ |m-m_1|> 1 \end{array}}{ z_0^k}(\varphi (m_1+z_1-m)-\varphi (-m))\\&=: I_0 + I_1 - I_2. \end{aligned}$$

Then, $R^{(m_1)}_{k}(\varphi )$ is holomorphic in a neighbourhood of $D(m_1)$.

We will first show that there is a constant $c_{1,k}$ such that

$$\begin{aligned} |R^{(m_1)}_{k}(\varphi )(z_0)|\le {c_{1,k}}\sup _{D_\infty }|\varphi | \quad \text { for all }z_0\in D(m_1). \end{aligned}$$

(6.29)

Note that this however is not an estimate of (6.27) for $n=1$ as the terms in the second line are all contained in the error term $R^{(m_1)}_{k}(\varphi )(z_0)$. We start by estimating $I_0$. Since $|z_0|\le 1$ and $I_0 = z_0^k\big (\varphi (-m_1-1)+\varphi (-m_1)+\varepsilon _1\varphi (-m_1+1)\big )$, we have

$$\begin{aligned} |I_0|\le 3 \sup _{D_\infty }|\varphi |. \end{aligned}$$

(6.30)

From (6.3) with $\{\omega :|\omega +m|=m-0.8\}\subset D_\infty$, we have

$$\begin{aligned} |\varphi ^{(j)}(-m)|\le {c_{k,j}'}\frac{j!\sup _{D_\infty }|\varphi |}{(m-0.8)^{k+j}}. \end{aligned}$$

Let ${c_{2,k}} = \sum _{m=1}^\infty \sum _{j=1}^k \frac{c_{k,j}'}{(m-0.8)^{k+j}}$. Since $\sum _{m=1}^\infty \frac{1}{(m-0.8)^{k+j}}<\infty$ for $k,j\ge 1,$

$$\begin{aligned} {c_{2,k}} = \sum _{m=1}^\infty \sum _{j=1}^k \frac{c_{k,j}'}{(m-0.8)^{k+j}} = \sum _{j=1}^k\sum _{m=1}^\infty \frac{c_{k,j}'}{(m-0.8)^{k+j}}< \infty . \end{aligned}$$

Then we have

$$\begin{aligned} |I_1| \le c_{2,k} \sup _{D_\infty }|\varphi |. \end{aligned}$$

(6.31)

Since $z_0\in D(m_1)$, we have $1/z_0 = z_1+m_1\in \Delta +m_1$ (see (4.7) for the definition of $\Delta$). Furthermore, $\frac{1}{m_1+1}\le |z_0| \le \frac{1}{m_1-1}$ if $m_1\not =1$ and $1/2\le |z_0| \le 1$ if $m_1=1$. Let us consider

$$\begin{aligned} I_2':= \sum _{\begin{array}{c} m \ge m_1 + 2 \end{array}}z_0^k (\varphi (1/z_0-m)-\varphi (-m)), \end{aligned}$$

and $I_2'':= I_2 - I_2'$. Then, $I_2'' = 0$ if $m_1\le 2$, and

$$\begin{aligned} I_2'' = \sum \limits _{1 \le m \le m_1- 2}z_0^k (\varphi (1/z_0-m)-\varphi (-m)) \quad \text {if }m_1\ge 3. \end{aligned}$$

If $m\ge m_1+2$, then $\text {Re}(1/{z_0}-m)\le -1$. For z in the segment $[-m, 1/{z_0}-m]\subset D_\infty$, the circle $\{\omega : |\omega -z|=m-0.8\}$ is contained in $D_\infty$. Thus, by (6.3), we have $|\varphi '(z)|\le c_{k,1}' \frac{\sup _{D_\infty }|\varphi |}{(m-0.8)^{k+1}}$ and thus

$$\begin{aligned} |I_2'| \le \sum _{m\ge m_1+2} \frac{c_{k,1}'}{(m-0.8)^{k+1}} \sup _{D_\infty }|\varphi |. \end{aligned}$$

If $m_1\ge 3$, then $1/z_0-m\in \Delta +(m_1-m)\subset D_\infty$ for $1\le m \le m_1-2$. Thus,

$$\begin{aligned} |I''_2|\le \frac{2(m_1-2)\sup _{D_\infty }|\varphi |}{(m_1-1)^k}\le 2\sup _{D_\infty }|\varphi |. \end{aligned}$$

Then, $|I_2|\le {c_{3,k}}\sup _{D_\infty }|\varphi |$, where ${c_{3,k}}:= \sum _{m=3}^\infty \frac{c_{k,1}'}{(m-0.8)^{k+1}} + 2.$ Hence, by combining the last estimate with (6.30) and (6.31) and setting $c_{1,k}:= c_{2,k}+c_{3,k}+3$, we obtain (6.29).

In the following we will estimate $T_k^n\varphi (z_0)$ itself which we split into a number of summands to be estimated separately. Recall $z_i$ as in (4.9) and (4.11). Iterating $T_k$ n times, we get

$$\begin{aligned} T_k^n\varphi (z_0)&=(\varphi (z_n)+\varphi (z_n-1) +\varepsilon _n\varphi (z_n+1))\prod _{i=0}^{n-1}(-z_i^k)\\&\qquad +\sum _{j=1}^{n-1}T_k^{n-j}\varphi (z_j-1)\prod _{i=0}^{j-1}(-z_i^k)\\&\qquad + \sum _{j=1}^{n-1}\varepsilon _jT_k^{n-j}\varphi (z_j+1)\prod _{i=0}^{j-1}(-z_i^k)\\&\qquad +\sum _{j=1}^nR^{(m_j)}_{k}(T_k^{n-j}\varphi )(z_{j-1})\prod _{i=0}^{j-2}(-z_i^k)\\&= : J_1+J_2+J_3+J_4, \end{aligned}$$

where $z_{-1}=1$. We recall (4.12) and (4.14), i.e., $\prod _{i=0}^{n-1}(-z_i^k) =(-1)^{n(k+1)} (p_{n-1}-q_{n-1}z_0)^{k}$ and $|p_{n-1}-q_{n-1}z_0|<q_{n}^{-1}$. Thus $J_1$ is the first term of (6.27). In the following steps (1)-(3) we estimate the summands $J_2$, $J_3$, and $J_4$.

• (1) We continue by estimating $J_4$. By using (4.12), (4.14) and (6.29) and by applying Proposition 6.3-(ii) to each of the summands and noting that $q_j\ge \left( \frac{\sqrt{5}-1}{2}\right) ^{-j}$, we obtain

  $$\begin{aligned} \left| \sum _{j=1}^nR^{(m_j)}_{k}(T_k^{n-j}\varphi )(z_{j-1})\prod _{i=0}^{j-2}(-z_i^k)\right|&\le \sum _{j=1}^n \frac{c_{1,k} \sup _{D_\infty } |T_k^{n-j}\varphi |}{q_{j-1}^k} \nonumber \\&\le c_{1,k} \sum _{j=1}^n\overline{C}_{0,k}\left( \frac{\sqrt{5}-1}{2}\right) ^{(n-1)k}\sup _{D_\infty } |\varphi | \nonumber \\&\le c_{4,k} \cdot n \left( \frac{\sqrt{5}-1}{2}\right) ^{nk}\sup _{D_\infty } |\varphi | \end{aligned}$$

  (6.32)

  with $c_{4,k} = c_{1,k} \overline{C}_{0,k}\left( \frac{\sqrt{5}+1}{2}\right) ^{k}.$

• (2) In the next steps, we will show that $|J_2|\le c_{5,k} n(\frac{\sqrt{5}-1}{2})^{nk}\sup _{D_\infty }|\varphi |$ for some $c_{5,k}$. If $z_j-1\not \in D_0$ for $1\le j < n$, then we have

  $$\begin{aligned} |T_k^{n-j}\varphi (z_j-1)| \le C_{K,k} \sup _{D_\infty } |(T_k^{n-j-1}\varphi )| \end{aligned}$$

  (6.33)

  by Proposition 6.1-(i) with $I=[0,1]$ and $K = \overline{(D - 1) \setminus D_0}$. On the other hand, let us consider the case $z_j-1\in D_0$. From (6.12), we deduce that

  $$\begin{aligned} |T_k^{n-j}\varphi (z_j-1)|&\le C_{I,k} |z_j-1|^k\log (1+|z_j-1|^{-1}) \sup _{D_\infty } |(T_k^{n-j-1}\varphi )| \\&+\left| \sum _{m =1}^\infty \sum _{i=1}^k \frac{(z_j-1)^{k-i}}{i!} (T_k^{n-j-1}\varphi )^{(i)}(-m)\right| . \end{aligned}$$

  From (6.31) and $|z_j-1|^k\log (1+|z_j-1|^{-1}) \le \log 2,$ we see that

  $$\begin{aligned} |T_k^{n-j}\varphi (z_j-1)| \le (C_{I,k} \log 2+c_{2,k})\sup _{D_\infty }|(T_k^{n-j-1}\varphi )|. \end{aligned}$$

  (6.34)

  Let $c_{6,k}= \max \{C_{K, k}, C_{I,k}\log 2+c_{2,k}\}$. Then, by combining the calculations leading to (6.32), (6.33) and (6.34), we get

  $$\begin{aligned} \left| \sum _{j=1}^{n-1}T_k^{n-j}\varphi (z_j-1)\prod _{i=0}^{j-1}(-z_i^k)\right| \le c_{5,k}\, n \left( \frac{\sqrt{5}-1}{2}\right) ^{nk}\sup _{D_\infty } |\varphi |, \end{aligned}$$

  where $c_{5,k} = \overline{C}_{0,k}\, c_{6,k} \left( \frac{\sqrt{5}+1}{2}\right) ^k$.

• (3) To complete the proof, it is enough to show that there is a constant $c_{7,k}$ such that

  $$\begin{aligned}&\Big |J_3 + \prod _{i=0}^{n-2}(-z_i^k)(1+z_{n-1})^k\varepsilon _{n-1}\varphi \Big (-\frac{z_{n-1}}{1+z_{n-1}}\Big )\Big | \le c_{7,k}\, n \left( \frac{\sqrt{5}-1}{2}\right) ^{nk}\sup _{D_\infty } |\varphi |. \end{aligned}$$

  (6.35)

  We have

  $$\begin{aligned}&J_3 + \prod _{i=0}^{n-2} (-z_i^k)(1+z_{n-1})^k\varepsilon _{n-1}\varphi \Big (-\frac{z_{n-1}}{1+z_{n-1}} \Big ) \\&= \sum _{j=1}^{n-2}\varepsilon _j T_k^{n-j}\varphi (z_j+1)\prod _{i=0}^{j-1} (-z_i^k) \\&\qquad + \left( T_k\varphi (z_{n-1}+1) + (1+z_{n-1})^k\varphi \left( -\frac{z_{n-1}}{1+z_{n-1}}\right) \right) \prod _{i=0}^{n-2}(-z_i^k) \\&=: I_4 + I _5. \end{aligned}$$

  • (3a) We will estimate $I_4$. If $m_{j+1}$ is small enough so that $z_j+1 \not \in D_1$, then by Proposition 6.1-(i) applied to $I=[0,1]$ and $K' = \overline{(D+1)\setminus D_1}$ we have

    $$\begin{aligned} |T_k^{n-j}\varphi (z_j+1)| \le C_{K',k} \sup _{D_\infty } |(T_k^{n-j-1}\varphi )|. \end{aligned}$$

    When $m_{j+1}$ is sufficiently large, then $z_j+1 \in D_1$. Then, by Proposition 6.8 and (6.31), we deduce that

    $$\begin{aligned}&|T_k^{n-j}\varphi (z_j+1)| \\&\le |1+z_{j}|^{k}\left| (T_k^{n-j-1}\varphi )\left( -\frac{z_{j}}{1+z_{j}}\right) \right| + C_k |z_{j}|\sup _{D_\infty }|T_k^{n-j-1}\varphi | \\&\qquad +\left| \sum _{m =1}^\infty \sum _{i=1}^k \frac{ (z_j+1)^{k-i}}{i!}(T_k^{n-j-1}\varphi )^{(i)}(-m)\right| \\&\le |1+z_j|^k\left| (T_k^{n-j-1}\varphi )\left( -\frac{z_j}{1+z_j}\right) \right| +(C_k+ c_{2,k} (2\sqrt{3}/3)^k)\sup _{D\infty }|T_k^{n-j-1}\varphi |. \end{aligned}$$

    Since $-\frac{z_j}{1+z_j}\in D_0$ and $j\le n-2$, by Proposition 6.7, we have

    $$\begin{aligned}&|1+z_j|^k\left| (T_k^{n-j-1}\varphi )\left( -\frac{z_j}{1+z_j}\right) \right| \\&\le C_{I,k} |1+z_j|^k \left| -\frac{z_j}{1+z_j}\right| ^k\log \left( 1+\left| -\frac{z_j}{1+z_j}\right| ^{-1}\right) \sup _{D_\infty }|T_k^{n-j-2}\varphi |\\&\qquad + |1+z_j|^k\left| \sum _{m =1}^\infty \sum _{i=1}^k \frac{(-z_j/(1+z_j))^{k-i}}{i!}(T_k^{n-j-2}\varphi )^{(i)}(-m)\right| . \end{aligned}$$

    Note that

    $$\begin{aligned}&|1+z_j|^k \left| -\frac{z_j}{1+z_j}\right| ^k\log \left( 1+\left| -\frac{z_j}{1+z_j}\right| ^{-1}\right) \\&= |z_j|^k\log \left( 1+\left| 1+\frac{1}{z_j}\right| \right) \le |z_j|^k \log \left( 2+\frac{1}{|z_j|}\right) . \end{aligned}$$

    Since $x^k\log (2+1/x)\rightarrow 0$ as $x\rightarrow 0$, it is increasing and $|z_j|<1$, it follows that $|z_j|^k\log (2+1/|z_j|)\le \log 3$. Thus, by a similar way to the proof of (6.35), it follows that

    $$\begin{aligned}{} & {} |1+z_j|^k\left| (T_k^{n-j-1}\varphi )\left( -\frac{z_j}{1+z_j}\right) \right| \le \left( C_{I,k} \log 3 + c_{2,k}\left( {2\sqrt{3}}/{3}\right) ^k\right) \\{} & {} \quad \sup _{D_\infty }|(T_k^{n-j-2}\varphi )|. \end{aligned}$$

    By letting $c_{8,k} = \max \big \{C_{K',k},~C_k+c_{2,k}\left( 2\sqrt{3}/3\right) ^k, C_{I,k}\log 3 + c_{2,k}$ $(2\sqrt{3}/3)^k\big \}$, we have

    $$\begin{aligned} \begin{aligned}&\left| I_4\right| \le c_{8,k} \sum _{j=1}^{n-2} \frac{\sup _{D_\infty }|T_k^{n-j-1}\varphi | + \sup _{D_\infty }|T_k^{n-j-2}\varphi |}{q_{j}^k}. \end{aligned} \end{aligned}$$

    (6.36)

  • (3b) We will estimate $I_5$. If $z_{n-1}+1\not \in D_1$, then $\text {Re}\left( \frac{1}{(z_{n-1}+1)}-1\right) <\frac{\sqrt{3}}{2}-1$. Thus, $\frac{1}{(z_{n-1}+1)}-1 \in D_\infty$. By Proposition 6.1-(i), we have

    $$\begin{aligned} \left| T_k\varphi (z_{n-1}+1) + (1+z_{n-1})^{k}\varphi \left( -\frac{z_{n-1}}{1+z_{n-1}}\right) \right| \le (C_{K',k}+2^k)\sup _{D_\infty }|\varphi |. \end{aligned}$$

    By Proposition 6.8 in the same manner as showing (6.31), we obtain that $z_{n-1}+1\in D_1$ implies

    $$\begin{aligned}&\left| T_k\varphi (z_{n-1}+1) + (1+z_{n-1})^{k}\varphi \left( -\frac{z_{n-1}}{1+z_{n-1}}\right) \right| \\&\le C_k |z_{n-1}|\sup _{D_\infty }|\varphi |+\left| \sum _{m =1}^\infty \sum _{i=1}^k \frac{(z_{n-1}+1)^{k-i}}{i!} \varphi ^{(i)}(-m)\right| \\&\le \big (C_k + c_{2,k}(2\sqrt{3}/3)^k \big ) \sup _{D_\infty }|\varphi |. \end{aligned}$$

    Let $c_{9,k} =\max \{C_{K',k}+2^k,~ C_k+c_{2,k}(2\sqrt{3}/3)^k\}$. Then, we have

    $$\begin{aligned} \left| I_5 \right| \le \frac{c_{9,k} \sup _{D_\infty }|\varphi |}{q_{n-1}^k}. \end{aligned}$$

    (6.37)

  Letting $c_{10,k} = \max \{c_{8,k},~ c_{9,k}\}$, from (6.36), (6.37) and (6.32), we obtain

  $$\begin{aligned} |I_4|+|I_5| \le 2 c_{10,k}\sum _{j=0}^{n-1} \frac{\sup _{D_{\infty }}|T_k^{n-j-1}\varphi |}{q_{j-1}^k} \le c_{7,k} n\left( \frac{\sqrt{5}-1}{2}\right) ^{nk}\sup _{D_\infty }|\varphi |, \end{aligned}$$

  where $c_{7,k} = 2 c_{10,k} \overline{C}_{0,k}(\frac{\sqrt{5}+1}{2})^{2k}$. Note that $c_{7,k}$ depends only on k.

$\square$

The next proposition is an analog of [26], Prop. 4.1]. We can show the following proposition in the same manner as in the proof in [26] by using (5.28) and Proposition 6.3-(2) for $k=1$ and the fact that $S=-T$.

Proposition 6.10

The following two statements hold true:

• (i) Let $I=[\gamma _0,\gamma _1]$, $\gamma _0>-1$. There exists $C_I>0$ such that for all $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus I)$ and for all $z\in D_0\cup H_0 \cup H_0'$, one has

  $$\begin{aligned} |S\varphi (z) + \sum _{m\ge 1}\varphi '(-m)|\le C_I |z|\log (1+|z|^{-1})\sup _U|\varphi |, \end{aligned}$$

  (6.38)

  where U is as in (6.11).

• (ii) There exists $C>0$ such that for all $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$ and for all $z\in D_0\cup H_0\cup H_0'$, one has

  $$\begin{aligned} |\overline{\sum _{\mathcal {M}}}\varphi (z)-\varphi (z) + \sum _{m\ge 1}(\overline{\sum _{\mathcal {M}}}\varphi )'(-m)|\le C|z|(1+\log |z|^{-1})\sup _{D_\infty }|\varphi |. \end{aligned}$$

  (6.39)

Concerning the boundary behaviour of the sum leading to the complex Wilton function, we simply observe that since $S=-T$ Propositions 4.1 and 4.4 of [26] hold. The same is true for Proposition 4.11 of [26], concerning the action of $T=-S$ on the space of analytic functions on $\overline{\mathbb {C}}\setminus [0,1]$ with bounded real part. However, the Wilton function will not have a bounded real part we thus omit its discussion.

7 Complex k-Brjuno and complex Wilton functions

In this section, we finally define the complex k-Brjuno and the complex Wilton functions. Let

$$\begin{aligned} \varphi _0(z):=-\frac{1}{\pi }\text {Li}_2\left( \frac{1}{z}\right) \quad \text { with }\quad \text {Li}_2\left( z\right) :=\sum _{n=1}^{\infty }\frac{z^n}{n^2}. \end{aligned}$$

(7.1)

Note that $\varphi _0\in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$, $\text {Im}\varphi _{0}(x)=0$ if $x\in \mathbb {R}\setminus [0,1]$ and

$$\begin{aligned} \text {Im}\varphi _0 (x\pm i0)=\pm \log \left( \frac{1}{x}\right) \quad \text {for}~ x\in (0,1]. \end{aligned}$$

(7.2)

We define $\tau := \big ({\begin{matrix}1&{}1\\ 0&{}1\end{matrix}}\big )$,

$$\begin{aligned} \varphi _{1, k}:= \left( L_{k,g(1)}+L_{k,\tau }\right) \varphi _0 \quad \text { and } \quad \overline{\varphi }_1:= \left( \overline{L}_{g(1)} + \overline{L}_{\tau }\right) \varphi _0. \end{aligned}$$

Note that $\overline{\varphi }_1 = (-L_{1,g(1)}+L_{1,\tau })\varphi _0$. Further, we note that $\varphi _0\in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$ implies $\varphi _{1,k},~ \overline{\varphi }_1 \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [1/2,2])$.

Similarly as in [26, Section 5] we note that from (6.8) we have that $\sum _{\mathcal {M},k}\varphi _{ 1, k} = \varphi _{1,k}+\sum _{\mathcal {M},k}T_k\varphi _{ 1, k}$. Since $Z\mathcal {M}= Z\mathcal {M}g(1) \sqcup Z\mathcal {M}\tau ,$ we have

$$\begin{aligned} \sum _{Z\mathcal {M},k}\varphi _0(z)=\sum _{Z,k}[\varphi _{1, k}(z)+\sum _{\mathcal {M},k}(T_k\varphi _{1, k}(z))] = \sum _{Z\mathcal {M},k}\varphi _{1,k}(z). \end{aligned}$$

(7.3)

Similarly, we have $\overline{\sum _{Z\mathcal {M}}}\varphi _0(z)=\overline{\sum _{Z\mathcal {M}}} \overline{\varphi }_1(z)$. We then deduce by simply applying the definitions of $L_{k,g}$ and $\overline{L}_g$

$$\begin{aligned} \varphi _{1, k}(z)=&-z^k \varphi _0\left( \frac{1}{z}-1\right) + \varphi _0(z-1) + \sum _{n=0}^k\frac{z^{k-n}}{n!}\varphi _0^{(n)}(-1) \end{aligned}$$

and

$$\begin{aligned} \overline{\varphi }_1(z) = z\left( \varphi _0\left( \frac{1}{z}-1\right) -\varphi _0(-1)\right) + \varphi _0(z-1) -\varphi _0'(-1). \end{aligned}$$

(7.4)

We deduce that

$$\begin{aligned} \varphi _{1,k}(z) = \frac{1}{\pi }\left[ {z^k}\text {Li}_2\left( \frac{z}{1-z}\right) - \text {Li}_2\left( \frac{1}{z-1}\right) \right] +\sum _{n=0}^k\frac{z^{k-n}}{n!}\varphi _0^{(n)}(-1), \end{aligned}$$

(7.5)

and since we additionally have $\varphi _0(-1) = \frac{\pi }{12}$ and $\varphi _0'(-1) = \frac{\log 2}{\pi }$, we also have

$$\begin{aligned} \overline{\varphi }_1(z) = -\frac{1}{\pi }\left[ z\text {Li}_2\left( \frac{z}{1-z}\right) +\text {Li}_2\left( \frac{1}{z-1}\right) \right] -z\,\frac{\pi }{12}-\frac{\log 2}{\pi }. \end{aligned}$$

(7.6)

By (7.2), we have

$$\begin{aligned} \text {Im}\varphi _{1,k}(x \pm i0) = {\left\{ \begin{array}{ll} \pm x^k\log \frac{x}{1-x},&{}\text {if~}1/2\le x<1, \\ \pm \log \frac{1}{x-1},&{}\text {if~}1<x\le 2 \end{array}\right. } \end{aligned}$$

(7.7)

and

$$\begin{aligned} \text {Im}\overline{\varphi }_1(x \pm i0)= {\left\{ \begin{array}{ll} \mp x\log \frac{x}{1-x} &{} \text {if}~1/2\le x<1,\\ \pm \log \frac{1}{x-1} &{} \text {if}~1<x\le 2. \end{array}\right. } \end{aligned}$$

(7.8)

Since $T_k$ is defined exactly in such a way that $T_k\varphi (x)=x^k\, \varphi (1/x)$ as an operator on the space of real functions and the k-Brjuno function fulfills the recursion $B_k(x)=-\log x+x^k\, B_k(1/x)$, we have $\left( 1-T_k\right) B_k)(x)=-\log x$ and therefore

$$\begin{aligned} B_k(x)=((1-T_k)^{-1}f)(x) \quad \text { with } \quad f(x)=\sum _{n\in \mathbb {Z}}\text {Im}\varphi _0(x+i0-n). \end{aligned}$$

(7.9)

We note here that by the above considerations $f(x)=\sum _{n\in \mathbb {Z}}\text {Im}\varphi _0(x+i0-n)=\log \{x\}$.

Then, the complex analytic extension of the real k-Brjuno function is $(1-T_k)^{-1}\left( \sum _{Z}\varphi _0\right)$ and we have $(1-T_k)^{-1}\left( \sum _{Z}\varphi _0\right) = \sum _{Z\mathcal {M},k}\varphi _0$ by applying (5.13), here $T_k$ is the extended operator as in (5.9). Hence, it is natural to define the complex k-Brjuno functions by

$$\begin{aligned} \mathcal {B}_k:= \sum _{Z\mathcal {M},k}\varphi _0. \end{aligned}$$

By applying the above consideration to W instead of $B_k$ and S instead of $T_k$, we may define the complex analytic extension of W by

$$\begin{aligned} \mathcal {W}:= \overline{\sum _{Z\mathcal {M}}}\varphi _0, \end{aligned}$$

(7.10)

which we call the complex Wilton function.

There is a two-to-one correspondence between $Z\mathcal {M}$ and $\mathbb {Q}$. From (5.11) and (5.12), we obtain analogous formulations for $\mathcal {B}_k$ and $\mathcal {W}$ respectively such that

$$\begin{aligned} \mathcal {B}_k(z)&= - \sum _{p/q\in \mathbb {Q}} \det \begin{pmatrix}p'&{}p\\ q'&{}q\end{pmatrix}^{k+1}\nonumber \\&\qquad \Bigg \{\frac{1}{\pi } \left[ (p'-q'z)^k \text {Li}_2\left( \frac{p'-q'z}{qz-p}\right) - (q''z-p'')^k\text {Li}_2\left( \frac{p''-q''z}{qz-p}\right) \right] \nonumber \\&\qquad \quad + \sum _{n=0}^k \frac{1}{n!}\,\det \begin{pmatrix}p'&{}p\\ q'&{}q\end{pmatrix}^{n} \nonumber \\&\qquad \qquad \quad \Bigg [\frac{(p'-q'z)^{k-n}}{(q')^n}\,\varphi _0^{(n)}\left( -\frac{q}{q'}\right) - \frac{ (q''z-p'')^{k-n}}{(q'')^n}\,\varphi _0^{(n)}\left( -\frac{q}{q''}\right) \Bigg ] \Bigg \}, \end{aligned}$$

(7.11)

where $[\frac{p'}{q'},\frac{p''}{q''}]$ is the Farey interval such that $\frac{p}{q}=\frac{p'+p''}{q'+q''}$ (with the convention $p'=p-1$, $q'=1$, $p''=1$, $q''=0$ if $q=1$). To obtain a representation as in (1.10) we remember the definition of $\varphi _0$ from (7.1) and note that

$$\begin{aligned} \varphi _{0}'(z) = - \frac{1}{\pi } \cdot \frac{1}{z} \log \left( 1-\frac{1}{z}\right) . \end{aligned}$$

If we let $f(z) = \frac{1}{z}$ and $g(z) = \log \left( 1-\frac{1}{z}\right)$, then $\varphi _0'(z) = -\frac{1}{\pi }f(z)g(z)$ and

$$\begin{aligned} f^{(j)}(z) = \frac{(-1)^j j!}{z^{j+1}} \text { for }j\ge 0, \end{aligned}$$

and

$$\begin{aligned} g^{(j)}(z) = \frac{(-1)^{j-1}(j-1)!}{z^j}\left( \left( \frac{z}{z-1}\right) ^j-1\right) \text { for } j\ge 1 \end{aligned}$$

holds. Since

$$\begin{aligned} (f g)^{(n)} = \sum _{i=0}^n {n\atopwithdelims ()i} f^{(n-i)}g^{(i)}, \end{aligned}$$

we have, for $n\ge 1$,

$$\begin{aligned} \varphi _0^{(n)} (z)&= -\frac{1}{\pi }\sum _{i=0}^{n-1} {n-1 \atopwithdelims ()i}f^{(n-1-i)}g^{(i)} \\&= -\frac{1}{\pi }\Bigg [ \frac{(-1)^{n-1}(n-1)!}{z^n} \log \left( 1-\frac{1}{z}\right) \\&\qquad \qquad + \sum _{i=1}^{n-1}\frac{(n-1)!}{(n-i-1)! i!}\,\frac{(-1)^{n-1-i}(n-1-i)!}{z^{n-i}}\,\frac{(-1)^{i-1}(i-1)!}{z^i}\nonumber \\&\qquad \qquad \qquad \cdot \left( \left( \frac{z}{z-1}\right) ^i-1\right) \Bigg ] \\&= -\frac{1}{\pi }\Bigg [ \frac{(-1)^{n-1}(n-1)!}{z^n} \log \left( 1-\frac{1}{z}\right) \\&\quad + \frac{(-1)^{n}(n-1)!}{z^n}\sum _{i=1}^{n-1}\frac{1}{i}\left( \left( \frac{z}{z-1}\right) ^i-1\right) \Bigg ] \\&= \frac{(-1)^{n}(n-1)!}{\pi z^n} \Bigg [ \log \left( 1-\frac{1}{z}\right) -\sum _{i=1}^{n-1}\frac{1}{i}\left( \left( \frac{z}{z-1}\right) ^i-1\right) \Bigg ] \end{aligned}$$

and by (7.11) we obtain (1.10). For the complex Wilton function, we obtain by the same considerations as above the representation as in (1.11).

7.1 Behaviour of $\mathcal {B}_k$ and $\mathcal {W}$ at rational points

7.1.1 Behaviour of $\mathcal {B}_k$ at rational points

Let

$$\begin{aligned} R_k(z):= \sum _{n=0}^k\frac{z^{k-n}}{n!}\varphi _0^{(n)}(-1), \end{aligned}$$

which coincides with the last term in (7.5). Note that $R_k(1)\in \mathbb {R}$ is well defined for all $k\in \mathbb {N}$, as $\varphi _0 \in C^\infty$.

Lemma 7.1

The function $\varphi _{1,k}{(z)}+i\log (1-z)$ is continuous on $\overline{\mathbb {H}} = \mathbb {H}\cup \mathbb {R}\cup \{\infty \}$ and its value at 1 is $R_k(1)+\frac{\pi }{2}$.

Proof

Recall that for $t\in \mathbb {C}\setminus [0,+\infty )$ we have Euler’s functional equation

$$\begin{aligned} \text {Li}_2\left( t\right) +\text {Li}_2\left( \frac{1}{t}\right) =-\frac{1}{2}(\log (-t))^2-\frac{\pi ^2}{6}. \end{aligned}$$

(7.12)

If $z\notin [0,+\infty )$, then $\frac{z}{1-z}, \frac{1}{z-1} \notin [0,+\infty )$. Thus, applying (7.12) with $t=\frac{z}{1-z}$ and $t=\frac{1}{z-1}$, and substituting it into (7.5) gives

$$\begin{aligned} \varphi _{1,k}(z)=&R_k(z)- z^k\frac{\pi }{6} + \frac{\pi }{6} -\frac{1}{\pi }\left( z^k\text {Li}_2\left( \frac{1-z}{z}\right) - \text {Li}_2\left( z-1\right) \right) \\&-\frac{1}{2\pi }\left( z^k\log ^2\left( \frac{- z}{1-z}\right) - \log ^2\left( \frac{1}{1-z}\right) \right) . \end{aligned}$$

The function $z^k\text {Li}_2\left( \frac{1-z}{z}\right) - \text {Li}_2\left( z-1\right)$ is continuous on $\overline{\mathbb {H}}$ and it is in $O(|z-1|)$ in the neighbourhood of 1. We then observe that

$$\begin{aligned}&z^k\log ^2\left( \frac{-z}{1-z}\right) - \log ^2\left( \frac{1}{1-z}\right) \\&=( z^k-1)\log ^2\left( \frac{-z}{1-z}\right) +\log ^2\left( \frac{-z}{1-z}\right) -\log ^2\left( \frac{1}{1-z}\right) , \end{aligned}$$

and $( z^k-1)\log ^2\left( \frac{-z}{1-z}\right) = O\big (|1-z|\log \big (\frac{1}{|1-z|}\big )\big )$ in the neighbourhood of 1. Also, we have

$$\begin{aligned} \log ^2\left( \frac{- z}{1-z}\right) - \log ^2\left( \frac{1}{1-z}\right)&= \left( \log (- z)+\log \left( \frac{1}{1-z}\right) \right) ^2 - \log ^2\left( \frac{1}{1-z}\right) \\&= \log ^2(- z) + 2\log (- z)\log \left( \frac{1}{1-z}\right) . \end{aligned}$$

In the neighbourhood of 1 in $\overline{\mathbb {H}}$, we have $\log (-z)+i\pi =O(|z-1|)$. Thus,

$$\begin{aligned} \log ^2\left( \frac{-z}{1-z}\right) - \log ^2\left( \frac{1}{1-z}\right) =-\pi ^2 - 2i\pi \log \left( \frac{1}{1-z}\right) +O\left( |z-1|\, \log \left( |1-z|\right) \right) . \end{aligned}$$

Therefore, in the neighbourhood of 1, we have

$$\begin{aligned} \varphi _{1,k}(z)= R_k(z) + \frac{\pi }{2} +i\log \left( \frac{1}{1-z}\right) +O\left( |z-1|\log \left( |1-z|\right) \right) \end{aligned}$$

(7.13)

and the result follows. $\square$

Corollary 7.2

The real part of $\varphi _{1,k}$ is bounded in $\overline{\mathbb {C}}\setminus \left[ 1/2,2\right]$. It has an extension to a continuous function on $\overline{\mathbb {C}}\setminus \{1\}$ and

$$\begin{aligned} \lim _{x\rightarrow 1^\pm } \text {Re}(\varphi _{1,k}(x))=R_k(1)\mp \frac{\pi }{2}. \end{aligned}$$

Proof

By (7.13), we have

$$\begin{aligned} \lim _{x\rightarrow 1^\pm } \text {Re}(\varphi _{1,k}(x))&=R_k(1)+\frac{\pi }{2}+\lim _{x\rightarrow 1^\pm } \text {Re}\left( i\log \left( \frac{1}{1- x}\right) \right) \\&=R_k(1)+\frac{\pi }{2}-\lim _{x\rightarrow 1^\pm } \arg \left( \frac{1}{1- x}\right) . \end{aligned}$$

The result follows from the fact that

$$\begin{aligned}&\lim _{x\rightarrow 1^+} \arg \left( \frac{1}{1- x}\right) =\pi \quad \text { and } \quad \lim _{x\rightarrow 1^-} \arg \left( \frac{1}{1- x}\right) =0. \end{aligned}$$

$\square$

Theorem 7.3

The real part of the complex k-Brjuno function has a decreasing jump of $\frac{\pi }{q^k}$ at each rational number $\frac{p}{q}$.

Proof

The proof follows similar arguments as that in [26, Section 5.2]. The space $\widehat{{\mathbb {H}}/\mathbb {Z}}$ is defined by

$$\begin{aligned} \widehat{{\mathbb {H}}/\mathbb {Z}}=\mathbb {H}/\mathbb {Z} \sqcup (\mathbb {R}\backslash \mathbb {Q})/\mathbb {Z}\sqcup \left( \overline{\mathbb {Q}/\mathbb {Z}}\times \left[ -\frac{\pi }{2}, +\frac{\pi }{2}\right] \right) , \end{aligned}$$

where $\overline{\mathbb {Q}/\mathbb {Z}}=\mathbb {Q}/\mathbb {Z}\cup \{\infty \}$. The space corresponds to a compactification of $(\overline{\mathbb {H}}\backslash \mathbb {Q})/\mathbb {Z}$ by attaching a semicircle on each $q\in \mathbb {Q}\cup \{\infty \}$. Then the value of a function on $\overline{\mathbb {Q}/\mathbb {Z}}\times [-\pi /2,\pi /2]$ is defined by

$$\begin{aligned} \varphi (\alpha ,\theta ):= \lim _{{\begin{array}{c} z\rightarrow \alpha \\ z\in \ell \end{array}}}\varphi (z), \end{aligned}$$

where $\ell$ is a ray emitting from $\alpha$ and the angle from $i\mathbb {R}$ to $\ell$ in the clockwise direction is $\theta$.

We note that the topology induced by $\widehat{\mathbb {H}/\mathbb {Z}}$ is the same as the topology induced by $\mathbb {H}/\mathbb {Z}\sqcup (\mathbb {R}\backslash \mathbb {Q})/\mathbb {Z}$. This implies the continuity of $\textrm{Re}\sum _{Z\mathcal {M},k} \varphi _0$ on $\widehat{{\mathbb {H}}/\mathbb {Z}}$ and hence the real part $\textrm{Re}\sum _{Z\mathcal {M},k} \varphi _0$ of the complex k-Brjuno function is continuous on ${\mathbb {H}}/\mathbb {Z}\sqcup (\mathbb {R}\backslash \mathbb {Q})/\mathbb {Z}$ in the usual sense. For $(\alpha _0,\pi /2)\in \overline{\mathbb {Q}/\mathbb {Z}}\times [-\pi /2,\pi /2]$, the value $\textrm{Re}\sum _{Z\mathcal {M},k}\varphi _0 (\alpha _0, \pi /2)$ (resp. $(\alpha _0, -\pi /2)$), with $\alpha _0 \in \mathbb {Q}/\mathbb {Z}$, is the right (resp. left) limit of $\textrm{Re}\sum _{Z\mathcal {M},k}\varphi _0 (\alpha )$, as $\alpha \in (\mathbb {R} \backslash \mathbb {Q})/\mathbb {Z}$ tends to $\alpha _0$. Recalling Corollary 7.2, one has

$$\begin{aligned} \textrm{Re} \varphi _{1,k} (1, \pi /2) - \textrm{Re} \varphi _{1,k} (1, -\pi /2) = -\pi \end{aligned}$$

and more precisely, by a similar argument as in the proof of Corollary 7.2, from (7.13), we have

$$\begin{aligned}&\text {Re}\varphi _{1,k}(1,\theta )-\text {Re}\varphi _{1,k}(1,0)\\&= - \lim _{t\rightarrow 0^+}\text {arg}\left( \frac{1}{1-(1+t\textrm{e}^{i( \pi /2 - \theta )})}\right) + \lim _{t\rightarrow 0^+}\text {arg}\left( \frac{1}{1-(1+t\textrm{e}^{i(\pi /2)})}\right) \\&= - \lim _{t \rightarrow 0^+}\text {arg}\left( t^{ -1} \textrm{e}^{i( \pi /2 + \theta )}\right) + \lim _{t\rightarrow 0^+}\text {arg}\left( t^{ -1}\textrm{e}^{i(\pi /2)}\right) =-\theta . \end{aligned}$$

If $\alpha _0\in \mathbb {Q}$, $\alpha _0\ne 1$, then $\textrm{Re}\varphi _{1,k} (\alpha _0, \theta ) = \textrm{Re} \varphi _{1,k} (\alpha _0, 0)$ for all $\theta \in [-\pi /2, \pi /2]$. Thus, by (7.3) one obtains that for all $p/q \in \mathbb {Q}$, $(p \wedge q = 1)$

$$\begin{aligned} \text {Re}\sum _{Z\mathcal {M}, k}\varphi _0(p/q, \theta ) = \textrm{Re} \sum _{Z\mathcal {M}, k} \varphi _0 (p/q, 0) - \theta /{q^k}. \end{aligned}$$

(7.14)

To show (7.14), we use the fact that

$$\begin{aligned} \text {Re}(L_{k,g}\varphi (\alpha ,\theta ))&= \det (g)^{k+1}(a-c\alpha )^k\left[ \text {Re}\varphi \left( \frac{d\alpha -b}{a-c\alpha }, \det (g)\theta \right) -\varphi \left( -\frac{d}{c}\right) \right] \nonumber \\&\qquad -\sum _{n=1}^k\frac{\det (g)^{k-n+1} (a-c\alpha )^{k-n}}{c^nn!}\varphi ^{(n)}\left( -\frac{d}{c}\right) . \end{aligned}$$

(7.15)

To verify this formula we note that for any $m\in \mathbb {N}$ we have

$$\begin{aligned}&\lim _{t\rightarrow 0+}\arg \left( \begin{pmatrix} 0&{}1\\ 1&{}m\end{pmatrix} .\left( \alpha + t\cdot e^{i(\pi /2-\theta )}\right) -\begin{pmatrix} 0&{}1\\ 1&{}m\end{pmatrix} .\alpha \right) \\&=\lim _{t\rightarrow 0+}\arg \left( -\frac{t\cdot e^{i(\pi /2-\theta )}}{(\alpha +m)(\alpha +m+t\cdot e^{i(\pi /2-\theta ))}}\right) =-\theta -\frac{\pi }{2}. \end{aligned}$$

(This is the ray we get after applying the Möbius transform.) Now, if $\varphi$ is analytic on the upper half-plane and real on the real line, we can apply Schwarz’ reflection principle and obtain

$$\begin{aligned} \text {Re}\varphi (\alpha , \theta )&= \lim _{t\rightarrow 0+} \text {Re}\varphi (\alpha + t\cdot e^{i(\pi /2-\theta )}) = \lim _{t\rightarrow 0+} \text {Re}\overline{\varphi }(\alpha + t\cdot e^{-i(\pi /2-\theta )})\\&= \lim _{t\rightarrow 0+} \text {Re}\varphi (\alpha + t\cdot e^{-i(\pi /2-\theta )}). \end{aligned}$$

Hence, if $\psi (z)=\varphi (g(m).z)$, then $\text {Re}\psi (\alpha , \theta )= \text {Re}\varphi (g(m).\alpha ,-\theta )$ and generally if $\psi (z)=\varphi (g.z)$ with $g\in \mathcal {M}$, then $\text {Re}\psi (\alpha , \theta )= \text {Re}\varphi (g.\alpha ,\det (g)\theta )$.

Since every element of $\mathcal {M}$ can be written as a product of matrices g(m), (7.15) follows.

By (7.3), we have

$$\begin{aligned}&\text {Re}\sum _{\mathcal{Z}\mathcal{M}, k}\varphi _0(\alpha ,\theta ) - \text {Re}\sum _{\mathcal{Z}\mathcal{M}, k}\varphi _0(\alpha ,0) \\&= \sum _{n\in \mathbb {Z},g\in \mathcal {M}}\text {Re}\left[ L_{k,g}\varphi _{1,k}(\alpha -n,\theta ) - L_{k,g}\varphi _{1,k}(\alpha -n,0) \right] . \end{aligned}$$

By Corollary 7.2, the real part $\text {Re}\big [L_{k,g}\varphi _{1,k}(\alpha -n,\theta )-L_{k,g}\varphi _{1,k}(\alpha -n,0)\big ]$ is 0 if and only if $\frac{d(\alpha -n)-b}{a-c(\alpha -n)} \not = 1$ and by (7.15),

$$\begin{aligned} \text {Re}\big [L_{k,g}\varphi _{1,k}(\alpha -n,\theta )-L_{k,g}\varphi _{1,k}(\alpha -n,0)\big ] = - \left( \det (g)(a-c(\alpha -n))\right) ^k\theta , \end{aligned}$$

if $\frac{d(\alpha -n)-b}{a-c(\alpha -n)}= 1$. There exist unique $n\in \mathbb {N}$ and $g =\big ({\begin{matrix}a&{}b\\ c&{}d \end{matrix}}\big ) \in \mathcal {M}$ such that $\frac{d(\alpha -n)-b}{a-c(\alpha -n)}= 1$, i.e. $\alpha = n + \frac{a+b}{c+d}.$ If $\alpha =p/q$, then $c+d=q$. Thus,

$$\begin{aligned} \text {Re}\sum _{\mathcal{Z}\mathcal{M}}\varphi _0(p/q,\theta ) - \text {Re}\sum _{\mathcal{Z}\mathcal{M}}\varphi _0(p/q,0) = - \frac{\text {det}(g)^{2k}}{(c+d)^k}\,\theta = -\frac{\theta }{q^k}. \end{aligned}$$

Thus, $\text {Re}(\mathcal {B}_k)$ has at each rational $p/q\in \mathbb {Q}/\mathbb {Z}$ a decreasing jump of $\pi /q^k$. $\square$

7.1.2 A comment on the Wilton function

It would be interesting to know the behaviour of the Wilton function at rational points. However, the results can not easily be transferred to the Wilton function. By a similar argument as in the proof of Lemma 7.1, we will get from the next lemma that $\overline{\varphi }_1(z)+i\log (1-z)-\frac{1}{2\pi }(z+1)\log ^2\left( 1-\frac{1}{z}\right)$ is continuous on $\overline{\mathbb {H}}$ and in a neighbourhood of 1 it behaves like

$$\begin{aligned} \overline{\varphi }_{1}(z){} & {} =\frac{3\pi }{4}-\frac{\log {2}}{\pi }-i\log \left( 1-z\right) +\frac{1}{2\pi }(z+1)\log ^2\left( 1-\frac{1}{z}\right) \\{} & {} \quad +O\left( |1-z|\log \left( |1-z|\right) \right) . \end{aligned}$$

Thus, we do not have an analog of Corollary 7.2 since $\text {Re}\big (\log ^2\big (1-\frac{1}{z}\big )\big )$ does not go to a finite value for $z\rightarrow 1$.

7.2 Behaviour of the imaginary part of $\mathcal {W}$

In this section, we will explain the behaviour of the imaginary part of the Wilton function near the real axis, distinguishing rational points, Wilton numbers and Diophantine numbers. Many arguments are slight adaptations of those given in [26], thus they will only be sketched and many proofs will be given in the Appendix.

The same behaviour which we observe for the imaginary part of the Wilton function will also characterize the approach to the real axis of the imaginary part of the k-Brjuno function, but we will limit ourselves to proving the results for the former, the arguments being very similar.

The following lemma is an analog to [26, Lem. 5.2] and Lemma 7.1. However, here we are considering the function $\overline{\varphi }_1(z)+i\log (1-z)-\frac{1}{2\pi }(z+1)\log ^2$ $\left( 1-\frac{1}{z}\right)$ instead of $\varphi _1(z)+i\log (1-z)$ in [26]. We will later remark on how this further influences the proof of Theorem 7.5.

Lemma 7.4

The function $\overline{\varphi }_1(z)+i\log (1-z)-\frac{1}{2\pi }(z+1)\log ^2\left( 1-\frac{1}{z}\right)$ is continuous on $\overline{\mathbb {H}}$ and its value at 1 is $\frac{3\pi }{4}-\frac{\log {2}}{\pi }$.

Proof

By combining (7.6) and Euler’s functional equation (7.12) it follows that

$$\begin{aligned} \begin{aligned} \overline{\varphi }_1(z) {=}&\frac{1}{\pi }\left[ z\text {Li}_2\left( \frac{1-z}{z}\right) {+}\text {Li}_2\left( z-1\right) \right] {+} \frac{1}{2\pi }\left[ z\log ^2\left( \frac{1-z}{-z}\right) +\log ^2(-z+1)\right] \\&+ \frac{\pi }{12}z+\frac{\pi }{6} - \frac{\log 2}{\pi }. \end{aligned}\nonumber \\ \end{aligned}$$

(7.16)

The function $z\text {Li}_2\left( \frac{1-z}{z}\right) +\text {Li}_2\left( z-1\right)$ is regular and vanishing at $z=1$. We have

$$\begin{aligned}{} & {} z\log ^2\left( \frac{1-z}{-z}\right) +\log ^2(-z+1) = (z+1)\log ^2\left( \frac{1-z}{-z}\right) \\{} & {} \quad +\log ^2(-z+1)-\log ^2\left( \frac{1-z}{-z}\right) . \end{aligned}$$

As in the proof of [26, Lem. 5.2], it follows that

$$\begin{aligned} \log ^2(-z+1)-\log ^2\left( \frac{1-z}{-z}\right) = \pi ^2 - 2\pi i \log (1-z) + O\left( |z-1|\log \left( |z-1|\right) \right) , \end{aligned}$$

which completes the proof. $\square$

From now on, we write $f(z) \lesssim g(z)$ if there exists a constant $C>0$ such that $f(z) \le C g(z)$ for all z. The following is an analog to [26, Thm. 5.10].

Theorem 7.5

For $n\ge 0$, $m_1,\cdots ,m_n\ge 1$ and $z_0\in H(m_1,\cdots ,m_n)$, we have

$$\begin{aligned} \text {Im}\mathcal {W}(z_0)=&- W_{\text {finite}}\left( \frac{p_n}{q_n}\right) +(-1)^{n}(p_{n-1}-q_{n-1}\text {Re}(z_0))\text {Im}\overline{\varphi }_1(z_n+1)\\&+r_n(z_0), \end{aligned}$$

with $|r_{n}(z_0)|\lesssim \frac{\log q_n}{q_n}|z_n|\log ^{2}(1+|z_n|^{-1}).$

As the proof is very similar to the one in [26, Thm. 5.11], it is given in the Appendix. However, the summand and the remainder term differ slightly from those in [26] about which we will comment in the following.

Remark 7.6

Here we give a remark on the summand $(-1)^n(p_{n-1}-q_{n-1}\text {Re}(z_0))$ $\text {Im}\overline{\varphi }_1(z_n+1)$ of Theorem 7.5. We have

$$\begin{aligned}{} & {} (-1)^n(p_{n-1}-q_{n-1}\text {Re}(z_0))\text {Im}\overline{\varphi }_1(z_n+1) \\{} & {} = \left( q_n^{-1}\log \frac{1}{|z_n|}\right) \left( 1+\frac{2}{\pi }\text {Arg}(1+1/z_n) + o(1) \right) . \end{aligned}$$

It can be derived as follows.

By (4.13), we have

$$\begin{aligned} q_n(p_{n-1}-q_{n-1}\text {Re}(z_0)) = (-1)^n \text {Re}\left( \frac{q_n}{q_{n-1}z_n+q_n}\right) \rightarrow (-1)^n \quad \text {as }z_n\rightarrow 0. \end{aligned}$$

By $\text {Re}(\log ^2(w)) = \log ^2|w|-(\text {Arg}(w))^2$ and $\text {Im}(\log ^2(x)) = 2\log |w|\text {Arg}(w)$, we have

$$\begin{aligned}&\text {Im}\left( (z_n+2)\log ^2\left( 1+\frac{1}{z_n}\right) \right) = \\&2(\text {Re}(z_n)+2)\text {Arg}(1+1/z_n)(\log |z_n+1|+\log |z_n|^{-1}) \\&+ \text {Im}(z_n)(\log ^2|1+1/z_n|-(\text {Arg}(1+1/z_n))^2). \end{aligned}$$

In the above equation, $2(\text {Re}(z_n)+2)\text {Arg}(1+1/z_n)\log |z_n+1|-\text {Im}(z_n)(\text {Arg}(1+1/z_n))^2$ is uniformly bounded. Since $|\text {Im}(z_n)\log ^2|1+1/z_n|\le |z_n|\log ^2(1+1/|z_n|)\rightarrow 0$ as $z_n\rightarrow 0$, from Lemma 7.4, we have

$$\begin{aligned} \frac{\text {Im}\overline{\varphi }_1(z_n+1)}{\left( \log \frac{1}{|z_n|}\right) }&=\frac{-\log |z_n| + \frac{1}{2\pi }2(\text {Re}(z_n)+2)\text {Arg}(1+1/z_n)\log \left| z_n\right| ^{-1}}{\log |z_n|^{-1}} + o(1) \\&=1+\frac{2}{\pi }\text {Arg}(1+1/z_n) + o(1) = 1 + O(1), \end{aligned}$$

but not in $1+o(1)$.

Finally, we remark here on the estimate of the remainder term in Theorem 7.5 which differs from the remainder term in [26, Thm. 5.10] as we have an additional factor of $\log q_n\cdot \log (1+|z_n|^{-1})$. It is caused by the signs of the terms $z\varphi _0(1/z-1)$ and $\varphi _0(z-1)$ in the definition of $\overline{\varphi }_1$ in (7.4). The functions $\varphi _0(1/z-1)$ and $\varphi _0(z-1)$ are not continuous at 1 because they can be written as a sum of a function continuous near 1 and a constant multiple of $\log ^2(1-z)$ (see the proof of Lemma 7.4). Unlike in the case of the Brjuno function, the non-continuous parts cancel out (see [26, Lem. 5.2]).

Finally, we will show an analog of [26, Thm. 5.19]. We will give a sufficient condition for W(x) to be approximated by the imaginary part of $\mathcal {W}(z)$ if z approaches x. For $R>0$ and $0< {r} < 1/2$, let

$$\begin{aligned} U_{R}{} & {} =\{u\in \mathbb {H}:~ \text {Im}(u) \ge |\text {Re}(u)|^{R}\}\quad \text {and}\quad \\ {\widetilde{U}}_{r}{} & {} = \left\{ u\in \mathbb {H}: ~\text {Im}(u) \ge \exp {\left( -\frac{1}{|\text {Re}(u)|^{r}} \right) } \right\} . \end{aligned}$$

Theorem 7.7

We have the following statements:

• (i) For any Wilton number x and any $R>0$, we have

  $$\begin{aligned} \lim _{u\rightarrow 0,~ u\in U_{R}} \text {Im}\mathcal {W}(u+x) = - W(x). \end{aligned}$$

• (ii) Let x be an irrational Diophantine number and $0< r < 1/2$ such that

  $$\begin{aligned} \liminf _{q\rightarrow \infty } \Vert q x \Vert _{\mathbb {Z}}~q^{1/r-1} = \infty , \end{aligned}$$

  where $\Vert \cdot \Vert _\mathbb {Z}$ denotes the distance from the nearest integer. Then,

  $$\begin{aligned} \lim _{u\rightarrow 0,~u\in {\widetilde{U}}_{r}} \text {Im}\mathcal {W}(u+x) = - W(x). \end{aligned}$$

Since the singular behaviour of the k-Brjuno function as the real line is approached is tamer than the singular behaviour of the Brjuno function, we expect that the same properties proven in [26] and stated in the theorem above for the Wilton function hold by a modest adaptation of the proofs given there.

Proof of Theorem 7.7

Let $x\in (0,1)$ be an irrational number whose continued fraction expansion is

$$\begin{aligned} x = \frac{1}{m_1+\frac{1}{m_2+\frac{1}{\ddots +\frac{1}{m_n +\ddots }}}}. \end{aligned}$$

Let $x_\ell = G^\ell (x)$. Let $\{p_n/q_n\}_{n\ge 0}$ be the sequence of the partial quotients of x. Let u be a point near 0 such that $|\text {Im}(u)|\le 1/2$ and $z = u + x \in \Delta$. Then there are $N_1 \cdots , N_L$ such that $z\in H(N_1,\cdots ,N_L)$. Let p/q be a rational number such that

$$\begin{aligned} p/q=\frac{1}{N_1+\frac{1}{N_2+ \frac{1}{\ddots +\frac{1}{N_L}}}}. \end{aligned}$$

We distinguish two cases.

• (i) Assume that $p/q = p_L/q_L$, i.e., $m_i = N_i$ for $i=1,\cdots , L$. From Theorem 7.5, Remark 7.6 and Lemma 3.1, there exist uniform constants $c_1,c_2,c_3>0$ such that

  $$\begin{aligned} \begin{aligned}&|\text {Im}\mathcal {W}(u+x) + W(x)| \\&=\Big |-W_{\textrm{finite}}\left( \frac{p_L}{q_L}\right) + \sum _{\ell =0}^{L-1}(-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell } + \sum _{\ell =L }^\infty (-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell } \\&\qquad +(-1)^n(p_{L-1}-q_{L-1}\text {Re}(z_0))\text {Im}\overline{\varphi }_1(z_L+1) +r_n(z_0) \Big | \\&\le c_1x_L q_n^{-1} + c_2q_{L}^{-1}\log |z_L|^{-1} + c_3 q_L^{-1} \log {(q_L)} |z_L|\log ^2(1+|z_L|^{-1}) \\&\qquad + \Big |\sum _{\ell =L}^\infty (-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell }\Big |. \end{aligned} \end{aligned}$$

  (7.17)

  This case follows in an analogous way to the proof of [26, Thm. 5.19(I)] where we note that the considered terms also stay small if we consider a Wilton instead of a Brjuno number.

• (ii) Assume that p/q is not one of the partial quotients of x. We denote by $\{p_\ell '/q_\ell '\}_{0\le \ell \le L}$ the partial quotient of p/q and by n the largest integer such that $p_n'/q_n'= p_n/q_n$. Clearly, one has $n<L$ and $p_L'/q_L' = p/q$. Note that $q_n^{-1}\log (q_n)\ge (q_L')^{-1}\log (q_L')$ if $q_n\ge 3$. As in the proof of [26, Thm. 5.19], from the fact $|x-p/q|\ge (2q^2)^{-1}$, if $u\in U_R$, then there exists a uniform constant $c>0$ such that

  $$\begin{aligned} q^{-1}\log |z_L|^{-1}\le q^{-1}(c+(2R-2)\log q) \le q_n^{-1}(c+(2R-2)\log q_n) \end{aligned}$$

  and if $u \in \widetilde{U_r}$, then we have

  $$\begin{aligned} q^{-1}\log |z_L|^{-1}\le q^{-1}cq^{2r}\le q_n^{-1}cq_n^{2r}. \end{aligned}$$

  From Theorem 7.5, Remark 7.6 and Lemma 3.1, there exist uniform constants $c_1,c_2,C_1,C_2,C_3>0$ such that

  $$\begin{aligned} \begin{aligned}&|\text {Im}\mathcal {W}(u+x) + W(x)| \\&=\Big |-W_{\textrm{finite}}\left( \frac{p_L'}{q_L'}\right) + \sum _{\ell =0}^{n-1}(-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell } + \sum _{\ell =n}^\infty (-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell }\\&\qquad +(-1)^L(p_{L-1}'-q_{L-1}'\text {Re}(z_0))\text {Im}\overline{\varphi }_1(z_L+1) +r_L(z_0) \Big | \\&\le \Big |-W_{\textrm{finite}}\left( \frac{p_L'}{q_L'}\right) + W_{\textrm{finite}}\left( \frac{p_n}{q_n}\right) \Big | + \Big | -W_{\textrm{finite}}\left( \frac{p_n}{q_n}\right) + \sum _{\ell =0}^{n-1}(-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell }\Big | \\&\qquad + \Big |\sum _{\ell =n}^\infty (-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell }\Big | + c_1\frac{\log |z_L|^{-1}}{q_{L}'} + c_2 \frac{\log q_L'}{q_L'} |z_L|\log ^2(1+|z_L|^{-1}) \\&\le \Big |-W_{\textrm{finite}}\left( \frac{p_L'}{q_L'}\right) + W_{\textrm{finite}}\left( \frac{p_n}{q_n}\right) \Big | + \Big |\sum _{\ell =n}^\infty (-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell }\Big | \\&\qquad + C_3 \frac{x_n}{q_n} + C_1 \frac{1}{q_n^{\gamma }} + C_2 \frac{\log q_n}{q_n} \end{aligned} \end{aligned}$$

  since $|z_L|\log ^2 (1+|z_L|^{-1})$ are bounded, where $\gamma = 1$ for the case of $u\in U_R$ and $\gamma = 1-2r$ for the case of $u\in {\widetilde{U}}_r$. Thus, it is enough to show that $W_\textrm{finite}(p/q)$ is close to $W_\textrm{finite}(p_n/q_n)$. Let

  $$\begin{aligned} \rho = \max _{n\le \ell < L} (q_\ell ')^{-1}\log \frac{q_{\ell +1}'}{q_\ell '}(\ell -n+1)^2. \end{aligned}$$

  By Lemma 3.1, we have

  $$\begin{aligned}&|W_{\textrm{finite}}(p_n/q_n) - W_{\textrm{finite}}(p/q)| \\&\le \left| W_{\textrm{finite}}(p_n/q_n) - \sum _{\ell =0}^{n-1}(-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell }\right. \\&\quad \left. + \sum _{\ell =0}^{L-1}(-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell } - W_{\textrm{finite}}(p/q)|\right| \\&\le C x_nq_n^{-1} + C x_L (q_L')^{-1} + \left| \sum _{\ell =n}^{L-1}(-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell }\right| \\&\le C' q_n^{-1} + \left| \sum _{\ell =n}^{L-1}(-1)^\ell \beta _{\ell -1}(x)\log \frac{1}{x_\ell }\right| . \end{aligned}$$

  We have

  $$\begin{aligned} \sum _{\ell =n}^{L-1} \beta _{\ell -1}(x)\log \frac{1}{x_\ell }\lesssim \sum _{\ell =n}^{L-1}\frac{1}{q_\ell '}\log \frac{q_{\ell +1}'}{q_\ell '} \le \rho \sum _{\ell =r}^{L-1}\frac{1}{(\ell -n+1)^2}\lesssim \rho . \end{aligned}$$

  With the same argument as in the proof of [26, Thm. 5.19] we have the conclusion.

Appendices

Appendix A. Convergence condition of ${\textsf{F}}_s$ for $1/2<s<1$

In this appendix, we show that the convergence of ${\textsf{F}}_s(x) = \sum _{n=1}^\infty \frac{e^{i\pi n^2 x}}{n^s}$ for $s\in (1/2,1)$ in [32] is equivalent to the convergence of a Brjuno-type function considered in [27]. Rivoal and Seuret proved the following:

Theorem 7.8

[32, Theorem 1-(i)] Let $x\in (0,1){\setminus } \mathbb {Q}$ and $s\in (1/2,1)$. If

$$\begin{aligned} \sum _{n=0}^\infty \frac{(q_{k+1}(x))^{(1-s)/2}}{(q_k(x))^{s/2}}<\infty , \end{aligned}$$

(A.1)

then ${\textsf{F}}_s(x)$ is absolutely convergent.

Let us consider the function $B_{\sigma ,\nu }$ for $\sigma >0$ and $\nu >0$ which satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} B_{\sigma ,\nu }(x+1) = B_{\sigma ,\nu }(x) &{} \text {for }x\in \mathbb {R}, \\ B_{\sigma ,\nu }(x) = {x^{-1/\sigma }} + x^\nu B_{\sigma ,\nu }(1/x) &{} \text {for }x\in (0,1). \end{array}\right. } \end{aligned}$$

(A.2)

When $\nu =1$, it can be obtained by replacing $-\log (x)$ by $x^{-1/\sigma }$ in (1.1) and it is recalled from [27, Eq. (3.6)] but here we consider a more general exponent $\nu >0$. Then, we have

$$\begin{aligned} B_{\sigma ,\nu }(x) = \sum _{n=0}^\infty (\beta _{n-1}(x))^\nu x_{n}^{-1/\sigma }. \end{aligned}$$

In the same way an operator T with $Tf(x)=x^{\nu }f(1/x)$ could be defined such that $(1-T)B_{\sigma ,\nu }(x)=x^{-1/\sigma }$. Furthermore, note that $x_n^{-1/\sigma } = \left( \frac{|p_n-q_nx|}{|p_{n-1}-q_{n-1}x|}\right) ^{-1/\sigma } \sim (q_n/q_{n+1})^{-1/\sigma }$ and $\beta _{n-1}(x) \sim 1/q_n$, here $f(x) \sim g(x)$ means that there is a constant $c>1$ such that $1/c<f(x)/g(x)<c$. Thus, $B_{\sigma ,\nu }(x)<\infty$ is equivalent to

$$\begin{aligned} \sum _{n=0}^\infty \frac{(q_{n+1}(x))^{1/\sigma }}{(q_n(x))^{\nu (1+1/\sigma )}} < \infty . \end{aligned}$$

If we take $\sigma = \frac{2}{1-s}$ and $\nu = \frac{s}{3-s},$ then we have the condition as in (A.1).

Appendix B. Proof of Proposition 6.3

Proof

From the fact that $\mathcal {M}$ is the free monoid generated by the matrices g(m), we have that $z\in D_\infty$ and $g\in \mathcal {M}$ imply $g^{-1}. z\in D_\infty$. Also, by this fact combined with [26, Lem. 3.2], we then have that

$$\begin{aligned} z\in V_\rho (D_\infty ) \text { implies } g^{-1}. z\in V_\rho (D_\infty ). \end{aligned}$$

(B.1)

Let $\mathcal {M}^{(r)}=\{g\in \mathcal {M}\;|\;g=g(m_1)\cdots g(m_r), ~ m_i\in \mathbb {N}\}$. Then we have for all $\psi \in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus I)$ that

$$\begin{aligned} (T_k^{(k+2)})^r\psi =\sum _{g\in \mathcal {M}^{(r)}}L^{(k+2)}_{g}\psi =\sum _{g'\in \mathcal {M}^{(r-1)}}L^{(k+2)}_{g'}(T_k^{(k+2)}\psi ). \end{aligned}$$

(B.2)

Let $g'=\left( {\begin{matrix} a'&{} b'\\ c'&{}d'\end{matrix}}\right)$, $z\in V_\rho (D_\infty )$ and $z'=(g')^{-1}. z$. Then for all $m\in \mathbb {N}$, we have

$$\begin{aligned} g'g(m)=\begin{pmatrix} b'&{} a'+mb'\\ d'&{}c'+md'\end{pmatrix}. \end{aligned}$$

Thus, by using the definition of $T_k^{(k+2)}$ given in Proposition 6.1 together with (5.26) and (5.27) we obtain

$$\begin{aligned} L^{(k+2)}_{g'}\left( T_k^{(k+2)}\psi \right)&=L^{(k+2)}_{g'}\left( \sum _{m=1}^\infty L_{g(m)}^{(k+2)}\psi \right) =L^{(k+2)}_{g'}\left( \sum _{m=1}^\infty -z^{-(k+2)}\psi \left( \frac{1}{z}-m\right) \right) \\&=\det (g)^{k+1}\left( a'-c'z\right) ^{-(k+2)} \left( -\sum _{m=1}^\infty \left( \frac{a'-c'z}{d'z-b'}\right) ^{k+2}\psi \left( \frac{1}{z'}-m\right) \right) \\&=-\det (g)^{k+1}(d'z-b')^{-(k+2)}\sum _{m=1}^\infty \psi \left( \frac{1}{z'}-m\right) . \end{aligned}$$

We remark that if $r=1$, then we consider $g' \in \mathcal {M}^{(0)} = \{\text {id}\}$ and the following arguments work well.

It follows from (5.1) and (B.1) that

$$\begin{aligned} \sum _{m=1}^\infty \left| \psi \left( \frac{1}{z'}-m\right) \right| \le \sum _{m=1}^\infty c_{1,\rho , k}\left| \frac{1}{z'}-m\right| ^{-(k+2)}\sup _{V_\rho (D_\infty )}|\psi | \end{aligned}$$

for some constant $c_{1,\rho , k}$. Since $\sup _{z\in D_\infty }\text {Re}(1/z) = 1$ (see Figure 7b), there exists $c_2>0$ such that $|1/z-m|^{-1}\le c_2m^{-1}$ for all $z\in D_\infty$ and $m\ge 2$. Thus, a similar argument as the one showing (6.2) in the proof of Proposition 6.1 yields that for all $z'\in V_\rho (D_\infty )$

$$\begin{aligned} \sum _{m=1}^{\infty }\left| \psi \left( \frac{1}{z'}-m\right) \right| \le {c_{3,\rho ,k}}\sup _{ V_\rho (D_\infty )}|\psi | \end{aligned}$$

for some constant ${c_{3,\rho , k}}$.

Also, there exists $c_{4,\rho }>0$ such that $|z-\frac{b'}{d'}|^{-1}\le c_{4,\rho }$ for all $z\in V_\rho (D_\infty )$ and $\frac{b'}{d'}\in [0,1]$. Here, $d'$ can be considered as the denominator of the rth principal convergent of some number. Finally,

$$\begin{aligned} \min _{g'\in \mathcal {M}^{(r-1)}}d'\ge \frac{1}{2}\left( \frac{\sqrt{5}+1}{2}\right) ^{r-1}\text { and } \sum _{g'\in \mathcal {M}^{(r-1)}}d'^{-2}\le c_5, \end{aligned}$$

(B.3)

for some $c_5>0$, see for example [26, Prop. A1.1 in Appendix 1]. Furthermore, since $g'\in \mathcal {M}^{(r-1)}$, we have that $\left| \det (g')^{k+1}\right| =1$. Combining these results we obtain the statement of (i).

In the following, we will show (ii) for $r\ge 2$, and we will discuss the case $r=1$ separately. Starting in an analogous way as in the proof of (i), using (5.23), we obtain for $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus [0,1])$ that

$$\begin{aligned} L_{k,g'}\left( T_k\varphi \right)&=L_{k,g'}\left( \sum _{m=1}^\infty L_{k,g(m)}\varphi \right) \\&=\frac{(c')^{-(k+1)}\det (g')^{k+1}}{k!}\,(a'-c'z)^{-1}\\&\qquad \cdot \int _0^1(1-t)^k \sum _{m=1}^\infty L_{k,g(m)}^{(k+2)}\varphi ^{(k+1)}\left( -\frac{d'}{c'}+\frac{\det (g')t}{c'(a'-c'z)}\right) \textrm{d}t. \end{aligned}$$

We have

$$\begin{aligned} \xi _{z,t}:{} & {} =-\frac{d'}{c'}+\frac{\det (g')t}{c'(a'-c'z)} = -\frac{d'}{c'}\, t+\frac{\det (g')t}{c'(a'-c'z)}+\frac{d'}{c'}\, t-\frac{d'}{c'}\\{} & {} = tg'^{-1}.z - (1-t)\frac{d'}{c'}. \end{aligned}$$

If we denote $\iota (z):=1/z$, see Fig. 7b, then we can inductively show that $g'^{-1}.z$ is contained in a $\rho$-neighbourhood of $\bigcup _{m\ge 1}(\iota (D_\infty )-m)\subset D_\infty$ with respect to the Poincaré metric. Since $d'/c' \in (-\infty ,-1]$, the line segment between $-d'/c'$ and $g^{-1}.z$ is contained in $V_\rho (D_\infty \cap \bigcup _{m\ge 1}(\iota (D_\infty )-m))$.

Thus, there is a constant $c_{6,\rho }>0$ such that $|\xi _{z,t}|\ge c_{6,\rho }$. The constant $c_{6,\rho }$ increases when $\rho \rightarrow 0$. Since $\xi _{z,t}$ is in a $\rho$-neighbourhood of i, $-i$, or $\text {Re}(\xi _{z,t})\le 0$, we have $|1/\xi _{z,t}-m|\ge m$.

Since $\varphi ^{(k+1)}\in \mathcal {O}^{k+2}(\overline{\mathbb {C}}\setminus [0,1])$, by using (5.26) and (6.3), we have

$$\begin{aligned} \sup _{z\in V_\rho (D_\infty )}|L_{k,g'}(T_k\varphi )|&\le \frac{(c')^{-(k+2)}}{k! \inf \limits _{z\in V_\rho (D_\infty )}|z-\frac{a'}{c'}|}\sum _{m\ge 1}\int _0^1\frac{|\varphi ^{(k+1)}\left( 1/\xi _{z,t}-m\right) |}{\left| \xi _{z,t} \right| ^{k+2}}dt\\&\le \frac{(c')^{-(k+2)}}{k! \inf \limits _{z\in V_\rho (D_\infty )}|z-\frac{a'}{c'}|}\sum _{m\ge 1}\frac{c_{1,k+1}'(k+1)! }{c_{6,\rho }^{k+2}(m-0.2)^{k+2}}\sup _{z\in V_\rho (D_\infty )}|\varphi (z)|\\&\le \frac{c_{7,k}c_{1,k+1}'(k+1)}{c_{6,\rho }^{k+2}} \cdot \frac{(c')^{-(k+2)}}{\inf \limits _{z\in V_\rho (D_\infty )}|z-\frac{a'}{c'}|} \cdot \sup _{z\in V_\rho (D_\infty )}|\varphi (z)|, \end{aligned}$$

where $c_{7,k} = \sum _{m\ge 1} (m-0.2)^{-(k+2)}$.

Since $a'/c'\in [0,1]$, we have $|z-\frac{a'}{c'}|^{-1}\le c_{4,\rho }$ for $z\in V_\rho (D_\infty )$. Using (B.3) and (6.3), we obtain

$$\begin{aligned} \sup _{z\in V_\rho (D_\infty )}|(T_k^r \varphi )(z)|&\le \frac{2^k c_{7,k} c_{1,k+1}'(k+1)}{c_{6,\rho }^{k+2}}\left( \frac{\sqrt{5}-1}{2}\right) ^{k(r-2)}\nonumber \\&\quad \sum _{g'\in \mathcal {M}^{(r-1)}}(c')^{-2} \cdot \sup _{z\in V_\rho (D_\infty )}|\varphi (z)|\nonumber \\&\le \overline{C}_{\rho ,k, \ge 2}\left( \frac{\sqrt{5}-1}{2}\right) ^{rk} \sup _{z\in V_\rho (D_\infty )}|\varphi (z)|, \end{aligned}$$

(B.4)

where $\overline{C}_{\rho ,k, \ge 2} = 2^k (k+1) (\frac{\sqrt{5}+1}{2})^{2k} c_{1,k+1}'c_5c_{4,\rho }c_{6,\rho }^{-(k+2)}c_{7,k}$, implying (ii) for $r\ge 2$.

For the case $r=1$, we will estimate

$$\begin{aligned} |T_k\varphi (z)| \le \sum _{m\ge 1} |L_{k,g(m)}\varphi (z)| \end{aligned}$$

by using (5.24) and (5.25). We consider $\ell$ denoting the right arc of $\partial D_\infty$ (which is the arc of the circle $\left| z-\frac{\sqrt{3}}{3}\right| = \frac{\sqrt{3}}{3}$ between $\frac{\sqrt{3}}{2}+\frac{1}{2}i$ and $\frac{\sqrt{3}}{2}-\frac{1}{2}i$ containing $\frac{2\sqrt{3}}{3}$), see Fig. 7a. Since $1/z\in V_\rho (\iota (D_\infty ))$, from the shape of $\iota (D_\infty )$, we can conclude that $t/z\in V_\rho (\iota (D_\infty ))$ for $0\le t\le 1$. Thus, $t/z-m\in V_\rho (\iota (D_\infty )-m)$. If $m\ge 2$ or z is not in the neighbourhood of $\ell$, then there is $c>0$ such that $|t/z-m|\ge cm$, thus by Cauchy’s estimate as in (6.3), we have

$$\begin{aligned} \left| \varphi ^{(k+1)}\left( \frac{t}{z}-m\right) \right| \le c_{1,k+1}'\frac{(k+1)!\sup _{V_\rho (D_\infty )}|\varphi |}{cm^{k+2}}. \end{aligned}$$

Thus, if z is not in the neighbourhood of $\ell$, then by using (5.25), we have

$$\begin{aligned} |T_k\varphi (z)|&\le \frac{1}{c_{6,\rho }k!} \sum _{m\ge 1} c_{1,k+1}'\frac{(k+1)!\sup _{V_\rho (D_\infty )}|\varphi |}{cm^{k+2}} \\&\le \overline{C}_{\rho ,k}' \left( \frac{\sqrt{5}-1}{2}\right) ^k \sup _{V_\rho (D_\infty )}|\varphi |,\nonumber \end{aligned}$$

(B.5)

where $\overline{C}_{\rho ,k}' = \left( \frac{\sqrt{5}-1}{2}\right) ^{-k}(k+1)c_{1,k+1}' c_{6,\rho }^{-1} c^{-1}\sum _{m\ge 1}\frac{1}{m^{k+2}}$. We are left to show the case that z is in a neighbourhood of $\ell$. For $z\in V_\rho (D_\infty )$ in a neighbourhood of $\ell$, we can assume that $|z|\le 2$. Thus, by (5.24) and (6.3), we have

$$\begin{aligned} |L_{k,g(1)}\varphi (z)|&\le |z|^k\left| \varphi \left( \frac{1}{z}-1\right) \right| + \sum _{n=0}^k\frac{|z|^{k-n}}{n!}|\varphi ^{(n)}(-1)| \nonumber \\&\le |z|^k \sup _{V_\rho (D_\infty )}|\varphi | + \sum _{n=0}^k \frac{|z|^{k-n}}{n!} c_{1,n}' \frac{n!\sup _{V_\rho (D_\infty )}|\varphi |}{0.8^{n+1}} \nonumber \\&\le \left( |z|^k + \sum _{n=1}^k \frac{c_{1,n}' |z|^{k-n}}{0.8^{n+1}} \right) \sup _{V_\rho (D_\infty )}|\varphi |\le C_k\left( \frac{\sqrt{5}-1}{2}\right) ^{k}\sup _{V_\rho (D_\infty )}|\varphi |, \end{aligned}$$

(B.6)

where $C_k = \left( \frac{\sqrt{5}-1}{2}\right) ^{-k}2^k \left( 1+ \sum _{n=1}^k \frac{c_{1,n}'}{2^n 0.8^{n+1}}\right)$. By replacing the summand of $m=1$ in (B.5) with (B.6) and combining this result with the estimate for $r\ge 2$ in (B.4), we have the conclusion when we take $\overline{C}_{\rho ,k} = \max \{C_k + \overline{C}_{\rho ,k}', \overline{C}_{\rho ,k,\ge 2}\}$. $\square$

Appendix C. Proof of Theorem 7.5.

Analogously as in [26] the strategy of the proof of the theorem is to start from the formula

$$\begin{aligned} \text {Im}\mathcal {W}(z)=\sum _{n\in {\mathbb {Z}}} \sum _{\ell \ge 0} \text {Im}S^\ell \overline{\varphi }_1(z+n). \end{aligned}$$

(C.1)

We will only give the parts of the proof which are different.

We have the following lemmas and propositions which are analogs to [26, Lem. 5.11, 5.14 and Prop. 5.15, 5.16] in the same way by using $S=-T$ and substituting $\overline{\varphi }_1$ for $\varphi _1$. For the following we denote for a bounded closed interval $I \subset \mathbb {R}$, the space of functions $\varphi \in \mathcal {O}^{1}(\overline{\mathbb {C}}\setminus I)$ whose real part is bounded, we denote it by E(I), endowed with the norm $\Vert \varphi \Vert _{E(I)} =\sup _{\mathbb {C}\backslash I} |\text {Re}\varphi |$. If it is clear which interval I is meant, we also write $\Vert \varphi \Vert$ instead of $\Vert \varphi \Vert _{E(I)}$. Note that $\overline{\varphi }_1\in E([1/2,2])$. If $\psi \in E([1/2,2])$ is real on the real axis outside [1/2, 2], then $\psi$ fulfils the arguments about $\varphi _1$ in the proof of [26, Lem. 5.11, 5.14 and Prop. 5.15, 5.16].

Lemma 7.9

For $z\in \Delta \setminus [0,1]$, if we write

$$\begin{aligned} \text {Im}\mathcal {W}(z)=&\text {Im}\overline{\varphi }_1(z)+\text {Im}\overline{\varphi }_1(z+1)+\text {Im}\overline{\varphi }_1(z+2) + \text {Im}S\overline{\varphi }_1(z-1) + \text {Im}S\overline{\varphi }_1(z) \\&+\sum _{\ell >1}\big [\text {Im}S^\ell \overline{\varphi }_1(z-1) + \text {Im}S^\ell \overline{\varphi }_1(z)+\text {Im}S^\ell \overline{\varphi }_1(z+1) \big ]+r^{0}(z), \end{aligned}$$

then we have

$$\begin{aligned} |r^{0}(z)|\le C|\text {Im}(z)|\Vert \overline{\varphi }_1\Vert _{E([1/2,2])}. \end{aligned}$$

Lemma 7.10

For $z\in \Delta \setminus [0,1]$, we have

$$\begin{aligned} \begin{aligned} \text {Im}\mathcal {W}(z) =&(1-\varepsilon _1)\text {Im}\overline{\varphi }_1(z) +\text {Im}\overline{\varphi }_1(z+1) +[1-\varepsilon _2(1-\varepsilon _1)]\text {Im}S \overline{\varphi }_1(z) \\&+\sum _{\ell >1}\text {Im}S^\ell \overline{\varphi }_1(z) + r^{0}(z) + r^{1}(z), \end{aligned}\end{aligned}$$

(C.2)

and

$$\begin{aligned} |r^{1}(z)|\le C|\text {Im}(z)|\log (1+|\text {Im}(z)|^{-1})\Vert \overline{\varphi }_1\Vert _{E([1/2,2])}. \end{aligned}$$

Proposition 7.11

For $n\ge 1$, $m_{1},\ldots ,m_{n}\ge 1$, $z_{0}\in D (m_{1},\ldots ,m_{n})$, $\ell >0$ we have

$$\begin{aligned}&\left| (-1)^{n}\text {Im}\left( S^{n+\ell }\overline{\varphi }_{1}(z_{0})\right) -\left[ \text {Im}\left( S^{\ell }\overline{\varphi }_{1}(z_{n})\right) \right] (p_{n-1}-q_{n-1}\text {Re}( z_{0}))\right| \\&\le Cq_{n}^{-1}|\text {Im}(z_{n})|\log (1+|\text {Im}(z_{n})|^{-1})\times {\left\{ \begin{array}{ll} \Vert S^{\ell -2}\overline{\varphi }_{1}\Vert &{}\text {if }\ell >1,\\ \Vert \overline{\varphi }_{1}\Vert &{}\text {if }\ell =1. \end{array}\right. } \end{aligned}$$

Proposition 7.12

For $n\ge 1$, $m_{1},\ldots ,m_{n}\ge 1$, $z_{0}\in D(m_{1},\ldots ,m_{n})$ we have

$$\begin{aligned}&|(-1)^{n}\text {Im}(S^n\overline{\varphi }_{1}(z_{0}))-(p_{n-1}-q_{n-1}\text {Re}( z_{0}))[ (1-\varepsilon _{n+1})\text {Im}(\overline{\varphi }_{1}(z_{n}))\\&\quad +\varepsilon _{n}\text {Im}( \overline{\varphi }_{1}(z_{n}+1))]|\le Cq_{n}^{-1}|\text {Im}( z_{n})|\log (1+|\text {Im}( z_{n}) |^{-1})\Vert \overline{\varphi }_{1}\Vert \; . \\\end{aligned}$$

The following is an analog of [26, Prop. 5.17]. We recall the definition of $H(m_1,\cdots ,m_n)$ given in (4.15).

Proposition 7.13

For $n \ge 0$, $m_{1},\ldots ,m_{n}\ge 1$, $z_0\in H(m_{1},\ldots ,m_{n})$ we have

$$\begin{aligned} \text {Im}\mathcal {W}(z_0)=&\sum _{\ell =0}^{n} {\varepsilon _{\ell }} {(-1)^\ell }(p_{\ell -1}-q_{\ell -1}\text {Re}(z_0)) \text {Im}\overline{\varphi }_1(z_\ell +1)\\&+\sum _{\ell =0}^{n-1}(1-\varepsilon _{\ell +1}) {(-1)^\ell }(p_{\ell -1}-q_{\ell -1}\text {Re}(z_0)) \text {Im}\overline{\varphi }_1(z_\ell ) + r_{n}(z_0), \end{aligned}$$

with $\varepsilon _0=1$, and $|r_{n}(z_0)|\le Cq_n^{-1}|z_n|\log (1+|z_n|^{-1})$.

Proof

Since $\overline{\varphi }_1$ is real on the real axis outside [0, 1], by Proposition 6.10, when $z\in H$,

$$\begin{aligned} |\text {Im}S^{n}\overline{\varphi }_1(z)|\lesssim |z|\log (1+|z|^{-1})\sup _{D_\infty }|S^{n-1}\overline{\varphi }_1|. \end{aligned}$$

(C.3)

We obtain $\sum _{n=1}^{\infty } \sup _{D_\infty }|S^{n} \overline{\varphi }_1|<\infty$ by using Proposition 6.3-(ii). Thus, from (C.3) and Lemma 7.10, we get

$$\begin{aligned} \text {Im}\mathcal {W}(z) = (1-\varepsilon _1)\text {Im}\overline{\varphi }_1(z) + \text {Im}\overline{\varphi }_1(z+1)+r(z), \end{aligned}$$

(C.4)

with $|r(z)|\lesssim |z|\log (1+|z|^{-1}).$

Let us consider the case $z_0\in H$ which corresponds to the case $n=0$. Since we set $\varepsilon _1=1$ for $z_0\in \Delta {\setminus } D$, from (C.4), we obtain

$$\begin{aligned} |\text {Im}\mathcal {W}(z_0)-\text {Im}\overline{\varphi }_1(z_0+1)|\lesssim |z_0|\log (1+|z_0|^{-1}). \end{aligned}$$

Next, we consider $z_0\in H(m_1,\cdots ,m_{n})$ with $n\ge 1$ and $m_1,\cdots ,m_{n}\ge 1$. By Lemma 7.10, we need to show that the modulus of

$$\begin{aligned} \begin{aligned}&\Big (\sum _{\ell =n+1}^\infty \text {Im}S^\ell \overline{\varphi }_1(z_0) \Big ) + (-1)^{n}(p_{n-1}-q_{n-1}\text {Re}(z_0))(1-\varepsilon _{n+1}) \text {Im}\overline{\varphi }_1(z_{n})+r^{0}(z_{0})+r^{1}(z_{0})\\&+ \Big [(1-\varepsilon _2(1-\varepsilon _1))\text {Im}S \overline{\varphi }_1(z_0) + (p_0-q_0\text {Re}(z_0))(\varepsilon _1\text {Im}\overline{\varphi }_1(z_1+1)+(1-\varepsilon _2)\text {Im}\overline{\varphi }_1(z_1))\Big ]\\&+ \sum _{\ell =2}^{n} \Big [\text {Im}S^\ell \overline{\varphi }_1(z_0) - (-1)^\ell (p_{\ell -1}-q_{\ell -1}\text {Re}(z_0))\big ( \varepsilon _{\ell }\text {Im}\overline{\varphi }_1(z_\ell +1) + (1-\varepsilon _{\ell +1})\text {Im}\overline{\varphi }_1(z_\ell )\big )\Big ] \\ \end{aligned}\nonumber \\ \end{aligned}$$

(C.5)

is in $O(q_{n}^{-1}|z_{n}|\log (1+|z_{n}|^{-1}))$. From [26, Prop. 4.11] using $S=-T$, we have $\sum _{j=0}^\infty \Vert S^j\overline{\varphi }_1\Vert <\infty$. Then, from Proposition 7.11, we have

$$\begin{aligned}&\Big |\Big (\sum _{\ell =n+1}^\infty \text {Im}S^\ell \overline{\varphi }_1(z_0) \Big )-(p_{n-1}-q_{n-1}\text {Re}(z_0))\sum _{\ell =1}^\infty \text {Im}S^\ell \overline{\varphi }_1(z_{n})\Big | \\&\lesssim q_{n}^{-1}|\text {Im}(z_{n})|\log (1+|\text {Im}(z_{n})|^{-1}). \end{aligned}$$

Since $z_{n}\in H$, by using (C.3), the first term of (C.5) is bounded by

$$\begin{aligned} \sum _{\ell =n+1}^\infty \big |\text {Im}S^\ell \overline{\varphi }_1(z_0) \big |\lesssim q_{n}^{-1}|z_{n}|\log (1+|z_{n}|^{-1}). \end{aligned}$$

For $z\in H$, we have

$$\begin{aligned} |\text {Im}\overline{\varphi }_1(z)|\lesssim |\text {Im}(z)|. \end{aligned}$$

(C.6)

For the second term of (C.5), by using (C.6) and (4.14), we have

$$\begin{aligned} |(p_{ n-1}-q_{n}-1\text {Re}(z_0))\text {Im}\overline{\varphi }_1(z_{n})(1-\varepsilon _{n+1})|\lesssim q_{n}^{-1}|\text {Im}(z_{n})|. \end{aligned}$$

We have $|r^{0}(z_0)+r^{1}(z_0)|\lesssim |\text {Im}(z_0)|\log (1+|\text {Im}(z_0)|^{-1})\Vert \overline{\varphi }_1\Vert _{E([1/2,2])}$ in Lemma 7.9 and 7.10. Since

$$\begin{aligned} {}[1-\varepsilon _2(1-\varepsilon _1)](1-\varepsilon _2)=1-\varepsilon _2\quad \text { and }\quad [1-\varepsilon _2(1-\varepsilon _1)]\varepsilon _1=\varepsilon _1, \end{aligned}$$

the third term of (C.5) is 0 or

$$\begin{aligned} \left[ \text {Im}S\overline{\varphi }_1(z_0)+(p_0-q_0\text {Re}(z_0))(\varepsilon _1\text {Im}\overline{\varphi }_1(z_1+1)+(1-\varepsilon _2)\text {Im}\overline{\varphi }_1(z_1))\right] . \end{aligned}$$

By using Proposition 7.12, the last terms of (C.5) are in

$$\begin{aligned} O\left( \sum _{\ell =0}^{n} q_\ell ^{-1}|\text {Im}(z_\ell )|\log (1+|\text {Im}(z_\ell )|^{-1})\Vert \overline{\varphi }_1\Vert _{E([1/2,2])}\right) \end{aligned}$$

and we have

$$\begin{aligned} \sum _{\ell =0}^{n} q_\ell ^{-1}|\text {Im}(z_\ell )|\log (1+|\text {Im}(z_\ell )|^{-1})\le q_{n}^{-1}|\text {Im}(z_{n})|\log (1+|\text {Im}(z_{n})|^{-1}) \end{aligned}$$

as in the last part of the proof of [26], Prop. 5.17]. Since $|\text {Re}(z_0)|\le 1$, we have the conclusion. $\square$

Proof of Theorem 7.5

Let $n\ge 0$ and $m_1,\cdots ,$ $m_n\ge 1$. The domain $H(m_1,\cdots ,m_n)$ meets $\mathbb {R}$ in a unique point $p_n/q_n$. Let us denote $x_0=p_n/q_n$ and consider its continued fraction

$$\begin{aligned} x_i^{-1}=m_{i+1}+x_{i+1},\quad 0\le i<n \quad \text {and} \quad x_n=0, \end{aligned}$$

here, $x_i$ is the point of intersection of $H(m_{i+1},\cdots ,m_n)$ with $\mathbb {R}$.

Let $z_0\in H(m_1,\cdots ,m_n)$ and $\text {Im}(z_0)>0$. We will then have $(-1)^\ell \text {Im}(z_\ell )>0$ for $0\le \ell \le n$. From (7.8), we deduce that

$$\begin{aligned} \text {Im}\overline{\varphi }_1 (x_\ell +1+(-1)^\ell i0) = (-1)^{\ell }\log \dfrac{1}{x_\ell }, \quad \text { for } ~ 0\le \ell <n. \end{aligned}$$

(C.7)

Since $\frac{x_\ell }{1-x_\ell }=\frac{1}{x_{\ell +1}}$ if $\varepsilon _{\ell +1}=0$, we obtain from (7.8)

$$\begin{aligned}{} & {} (1-\varepsilon _{\ell +1})\text {Im}\overline{\varphi }_1(x_\ell +(-1)^\ell i0) = (1-\varepsilon _{\ell +1})(-1)^{\ell +1} x_\ell \log \dfrac{1}{x_{\ell +1}},\nonumber \\{} & {} \quad \text { for }~0\le \ell < n-1. \end{aligned}$$

(C.8)

The above equations imply for $W_{\text {finite}}$ extended to the complex plane that

$$\begin{aligned} \begin{aligned} W_{\text {finite}}(p_n/q_n) =&\sum _{\ell =0}^{n-1}\varepsilon _\ell (q_{\ell -1}x_0-p_{\ell -1})(-1)^{\ell } \text {Im}\overline{\varphi }_1(x_\ell +1+(-1)^\ell i0) \\&+ \sum _{\ell =0}^{n-2} (1-\varepsilon _{\ell +1}) (q_{\ell -1} x_0-p_{\ell -1})(-1)^{\ell } \text {Im}\overline{\varphi }_1(x_\ell +(-1)^\ell i0). \end{aligned} \end{aligned}$$

(C.9)

Combining Proposition 7.13 with (C.9), we get

$$\begin{aligned} \begin{aligned}&\text {Im}\mathcal {W}(z_0)+W_{\text {finite}}(p_n/q_n)-(-1)^n(p_{n-1}-q_{n-1}\text {Re}(z_0))\text {Im}\overline{\varphi }_1(z_n+1) - r_n(z_0) \\&= \sum _{\ell =0}^{n-1} \varepsilon _{\ell } {(-1)^\ell } \Big [ (p_{\ell -1}-q_{\ell -1}\text {Re}(z_0)) \text {Im}\overline{\varphi }_1(z_\ell +1)\\&\quad - (p_{\ell -1}-q_{\ell -1}x_0)\text {Im}\overline{\varphi }_1(x_\ell +1+(-1)^\ell i0) \Big ] \\&+\sum _{\ell =0}^{n-2}(1-\varepsilon _{\ell +1}) {(-1)^\ell } \Big [ (p_{\ell -1}-q_{\ell -1}\text {Re}(z_0)) \text {Im}\overline{\varphi }_1(z_\ell )\\&\quad - (p_{\ell -1}-q_{\ell -1} x_0) \text {Im}\overline{\varphi }_1(x_\ell +(-1)^\ell i0) \Big ] \\&- (1-\varepsilon _n)(-1)^n\Big [ (p_{n-1}-q_{n-1}\text {Re}(z_0))\text {Im}\overline{\varphi }_1(z_{n}+1)\\&\quad + (p_{n-2}-q_{n-2}\text {Re}(z_0))\text {Im}(\overline{\varphi }_1(z_{n-1})) \Big ]. \end{aligned} \end{aligned}$$

(C.10)

Thus, we have to deal with the following expressions. For $0\le \ell \le n-1$ such that $\varepsilon _\ell =1$,

$$\begin{aligned} A_\ell := (p_{\ell -1}-q_{\ell -1}\text {Re}(z_0)) \text {Im}\overline{\varphi }_1(z_\ell +1) - (p_{\ell -1}-q_{\ell -1}x_0)\text {Im}\overline{\varphi }_1(x_\ell +1+(-1)^\ell i0). \end{aligned}$$

For $0\le \ell \le n-2$ such that $\varepsilon _{\ell +1}=0$,

$$\begin{aligned} B_\ell := (p_{\ell -1}-q_{\ell -1}\text {Re}(z_0)) \text {Im}\overline{\varphi }_1(z_\ell ) - (p_{\ell -1}-q_{\ell -1} x_0) \text {Im}\overline{\varphi }_1(x_\ell +(-1)^\ell i0). \end{aligned}$$

When $\varepsilon _n=0$,

$$\begin{aligned} C_n:= (p_{n-1}-q_{n-1}\text {Re}(z_0))\text {Im}\overline{\varphi }_1(z_{n}+1) + (p_{n-2}-q_{n-2}\text {Re}(z_0))\text {Im}\overline{\varphi }_1(z_{n-1}). \end{aligned}$$

The first n digits of the continued fraction of $\text {Re}(z_0)$ are also $m_1,\cdots ,m_n$. This gives

$$\begin{aligned} |\text {Re}(z_0)-x_0|\le |z_0 - p_n/q_n| = q_n^{-1}|q_{n-1}z_0-p_{n-1}||z_n| \le q_n^{-2}|z_n|. \end{aligned}$$

Since (7.6) implies that

$$\begin{aligned} \overline{\varphi }_1'(z) = \frac{1}{\pi }\left( - \text {Li}_2\left( \frac{z}{1-z}\right) + \frac{1}{1-z}\log \left( \frac{1-2z}{1-z}\right) + \frac{1}{1-z}\log \left( \frac{z-2}{z-1}\right) \right) - \frac{\pi }{12}, \end{aligned}$$

we have

$$\begin{aligned} |\overline{\varphi }_1'(z)|\le C|z-1|^{-1}\log \frac{1}{|z-1|} \quad \text {near }1. \end{aligned}$$

(C.11)

For $0\le \ell \le n$, the distance of $x_\ell$ and $z_\ell$ from 0 are comparable. Thus, for $0\le \ell \le n$,

$$\begin{aligned} |\text {Im}\overline{\varphi }_1(z_\ell +1)-\text {Im}\overline{\varphi }_1(x_\ell + 1 + (-1)^\ell i 0)|\le Cx_\ell ^{- 1}\log {\frac{1}{x_\ell }}|z_\ell - x_\ell |, \end{aligned}$$

and, for $0\le \ell \le n-1$,

$$\begin{aligned} |\text {Im}\overline{\varphi }_1(z_\ell )-\text {Im}\overline{\varphi }_1(x_\ell + (-1)^\ell i 0)|\le C|x_\ell -1|^{-1}\log \frac{1}{1-x_\ell }|z_\ell - x_\ell |. \end{aligned}$$

We have $|z_\ell - x_\ell |\lesssim |z_n|q_\ell ^2q_n^{-2}$, $x_\ell ^{-1}\lesssim m_{\ell +1}$ and $|x_{\ell }-1|^{-1}\lesssim x_{\ell +1}^{-1}\lesssim m_{\ell +2}$. Thus, according to (C.7) and (C.8), for $0\le \ell <n$, we have

$$\begin{aligned} |A_\ell |&= \Big |(p_{\ell -1}-q_{\ell -1}\text {Re}(z_0)) \Big (\text {Im}\overline{\varphi }_1(z_\ell +1) - \text {Im}\overline{\varphi }_1(x_\ell +1+(-1)^\ell i 0)\Big ) \\&\qquad + q_{\ell -1}(\text {Re}(z_0)- x_0)\text {Im}\overline{\varphi }_1(x_\ell +1+(-1)^\ell i0) \Big | \\&\lesssim x_\ell ^{-1}\log {\frac{1}{x_\ell }}|z_\ell -x_\ell |q_\ell ^{-1} + q_{\ell -1}q_n^{-2}|z_n|\log \frac{1}{x_\ell } \\&\lesssim |z_n|q_n^{-2}(q_\ell m_{\ell +1} + q_{\ell -1}) \log {(m_{\ell +1})} \lesssim |z_n|q_n^{-2}q_{\ell +1}\log {(m_{\ell +1})}, \end{aligned}$$

and, for $0\le \ell <n-1$ such that $\varepsilon _{\ell +1}=0$,

$$\begin{aligned} |B_\ell |&= \Big |(p_{\ell -1}-q_{\ell -1}\text {Re}(z_0)) \Big (\text {Im}\overline{\varphi }_1(z_\ell ) - \text {Im}\overline{\varphi }_1(x_\ell +(-1)^\ell i 0)\Big ) \\&\qquad + q_{\ell -1}(\text {Re}(z_0)- x_0)\text {Im}\overline{\varphi }_1(x_\ell +(-1)^\ell i0) \Big | \\&\lesssim | p_{\ell -1}-q_{\ell -1}\text {Re}(z_0) | |x_\ell -1|^{-1} \log \frac{1}{|x_\ell -1|} |z_\ell - x_\ell | + q_{\ell -1}q_n^{-2}|z_n| \log \frac{1}{x_{\ell +1}} \\&\lesssim |x_\ell -1|^{-1}\log \frac{1}{|x_\ell -1|}|z_\ell -x_\ell |q_\ell ^{-1} + q_{\ell -1}q_n^{-2}|z_n|\log \frac{1}{x_{\ell +1}} \\&\lesssim |z_n|q_n^{-2}(q_\ell m_{\ell +2} + q_{\ell -1}) \log (m_{\ell +2}) \lesssim |z_n|q_n^{-2}q_{\ell +2} \log (m_{\ell +2}). \end{aligned}$$

From $q_n\ge 2^{\lfloor (n-\ell )/2 \rfloor }q_{\ell }$, it follows that

$$\begin{aligned} \sum _{\ell =0}^{n-1}|A_\ell |+\sum _{\ell =0}^{n-2}|B_\ell | \lesssim |z_n|\,\frac{\log {q_n}}{q_n}. \end{aligned}$$

Furthermore, when $\varepsilon _n=0$, we have

$$\begin{aligned} \overline{\varphi }_1(z_{n-1}) = z_{n-1} \left( \varphi _0\left( z_n\right) -\varphi _0(-1)\right) -\varphi _0'(-1){+}\varphi _0(z_{n-1}-1).\\ \overline{\varphi }_1(1+z_{n}) = z_{n-1}^{-1} \left( \varphi _0\left( z_{n-1}-1\right) -\varphi _0(-1)\right) -\varphi _0'(-1){+}\varphi _0(z_{n}). \end{aligned}$$

Since

$$\begin{aligned} \overline{\varphi }_1(z_{n-1}) = (1-z_{n-1})\left( \frac{\pi }{12}-\frac{1}{\pi }\log 2\right) + z_{n-1}\overline{\varphi }_1(1+z_n), \end{aligned}$$

we have

$$\begin{aligned} C_n&= \text {Im}\overline{\varphi }_1(z_{n}+1) [(p_{n-1}-q_{n-1}\text {Re}(z_0)) + (p_{n-2}-q_{n-2}\text {Re}(z_0))\text {Re}(z_{n-1})] \\&+ (p_{n-2}-q_{n-2}\text {Re}(z_0))\text {Im}(z_{n-1})\left[ -\left( \frac{\pi }{12}-\frac{1}{\pi }\log 2\right) +\text {Re}\overline{\varphi }_1(1+z_n))\right] . \\ \end{aligned}$$

By (7.16), we have $\left| \overline{\varphi }_1(1+z_n))\right| \lesssim \log ^2(1+|z_n|^{-1}).$ As in the proof of [26, Theorem 5.10],

$$\begin{aligned} |(p_{n-2}-q_{n-2}\text {Re}(z_0))\text {Im}(z_{n-1})|\lesssim q_{n}^{-1}|\text {Im}(z_n)|, \end{aligned}$$

and

$$\begin{aligned}{}[(p_{n-1}-q_{n-1}\text {Re}(z_0)) + (p_{n-2}-q_{n-2}\text {Re}(z_0))\text {Re}(z_{n-1})] \lesssim q_{n}^{-1}|\text {Im}(z_n)|. \end{aligned}$$

Thus, we have $|C_{n}|\lesssim q_{n}^{-1}|\text {Im}(z_n)|\log ^2(1+|z_n|^{-1})$.