Regularity properties of $k$-Brjuno and Wilton functions

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.


Introduction
Let G denote the Gauss map, that is G(0) = 0 and G(x) = 1 x otherwise, where {x} = x − x and let for j ≥ 0 with the convention β −1 (x) = 1. This condition is called Brjuno condition and was introduced by Brjuno in the study of certain problems in dynamical systems, see [Brj71,Brj72]. 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 of C.L. Siegel [Sie42], Brjuno proved that all germs of holomorphic diffeomorphisms of one complex variable with an indifferent fixed point with linear e 2πix are linearisable if x is a Brjuno number. Conversely, in 1988 J.-C. Yoccoz [Yoc88,Yoc95] proved that this condition is also necessary. Similar results hold for the local conjugacy of analytic diffeomorphisms of the circle [Yoc02] and for some complex area-preserving maps [Mar90,Dav94]. This condition has been of interest also for in different contexts. For instance it is conjectured that the Brjuno condition is optimal for the existence of real analytic invariant circles in the standard family [Mac88,Mac89,MS92]. See also [BG01,Gen15] and references therein for related results.
Furthermore, the Brjuno function satisfies the functional equation for x ∈ (0, 1). The second author together with Moussa and Yoccoz investigated the regularity properties of B 1 in [MMY97] and later constructed an analytic extension of B 1 to the complex plane [MMY01]. Let T denote the linear operator and, as proven in [MMY97], by exploiting the fact that the operator T as in (1.1) 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 . Local properties of the Brjuno function have been recently investigated by M. Balazard and B. Martin [BM12] and its multifractal spectrum was determined by S. Jaffard and B. Martin in [JM18].
1.2. k-Brjuno functions. For the following, for k ≥ 2 even, let E k be the Eisenstein series of weight k defined in the upper-half plane H = {z ∈ C : Imz > 0}. Then its Fourier expansion is given by where B k is the kth Bernoulli number and σ k−1 (n) = d|n d k−1 . For all k ≥ 4, E k is modular of weight k under the action of SL 2 (Z), and E 2 is quasi-modular of weight 2 under the action of SL 2 (Z), see for example [Zag92]. The function E 2 can be viewed as a modular (or Eichler) integral on SL 2 (Z) of weight 2 with the rational period function − 2πi z , see for example [Kno90].
For k ≥ 2 even and z ∈ H, denote ϕ k (z) = ∞ n=1 σ k−1 (n) n k+1 e 2πinz . We have that Consider the imaginary part of ϕ k Analytic properties, differentiability and Hölder regularity exponent, of the function F k (and the real part of ϕ 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 ∞ n=0 diverges, then F 2 is not differentiable at x ∈ R \ Q. It has been conjectured that for all see [Pet14,Pet17]. The occurrence of a condition of this type motivates the following definition.
be called k-Brjuno function. From this equation we already get the implicit definition B k (x) = log(1/x) + x k · B k (G(x)). It converges at an irrational x if and only if (1.3) holds, see also Proposition 2.5 for an even stronger statement about the relation between the k-Brjuno function and (1.3). Obviously, for k = 1, the function in (1.4) gives the Brjuno function introduced before. Instead of considering the k-Brjuno function with respect to the Gauss map as in (1.4) it is also possible to use α-continued fractions instead. The classical Brjuno function associated to α-continued fractions was already investigated in [MMY97]. Let α ∈ 1 2 , 1 and let A α : (0, α) → [0, α] the transformation of the α-continued fractions being given by (1.5) For α = 1, we obtain the Gauss map associated to the regular continued fraction transformation and for α = 1/2, we obtain the transformation associated to the nearest integer continued fractions. Nakada [Nak81] was the first to consider all these types of continued fractions as a one-parameter family, however he considered the 'unfolded' version of the α-continued fraction which is defined by the Gauss map A α (x) = 1 x − 1 x − α + 1 . The version as in (1.5) was little later considered in [TI81].
−1 (x) = 1, be the generalisation of (1.4) in the sense that we consider the k-Brjuno function not only for the Gauss transformation but also for other α-continued fraction transformations A α with α ∈ [1/2, 1].
For given α ∈ [1/2, 1], any x ∈ (0, 1] has the α-continued fraction expansion given by , where ν corresponds to the exponent k in (1.6). It is understood that the function T k,α f is completed outside the interval (0, α) by imposing to T k,α f the same parity and periodicity conditions imposed to f . Then we have which follows by a simple calculation.
1.3. Wilton function. Next we consider the related concept of the Wilton function which is given by namely by the alternate signs version of the Brjuno function series (1.1). (see Figure 1 for its graph). It converges if and only if it fulfills the Wilton condition 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 [Wil33]. 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 ∈ (0, 1) being a Wilton number: which by using the same linear operator T as in (1.1) can be written as We can extend the Wilton function in the same way we did for the k-Brjuno functions, i.e. replacing the Gauss map with the transformation of the α-continued fractions: for α ∈ 1 2 , 1 and for all irrational x we define .
We then have where the operator S α = −T 1,α . Also in this case it is understood that S α acts on Z- The Wilton function and its primitive have been studied recently by Balazard and Martin in terms of its convergence properties [BM19] and in the context of the Nyman and Beurling criterion, see [BM12,BM13] and [BDBLS05].
1.4. Structure of the paper. The paper is organised as follows. In Section 2 we state and prove the BMO-properties of the k-Brjuno and the Wilton function and in Section 3 we give statements about the truncated k-Brjuno and the truncated Wilton function.

BMO properties of the real k-Brjuno and the real Wilton function
In this section, we study the bounded mean oscillation (BMO) properties of the real k-Brjuno and the Wilton function -both with respect to different transformations A α with α ∈ [1/2, 1]. Before stating the main results of this section, we will first recall the definition of a BMO function.
Let L 1 loc (R) be the space of the locally integrable functions on R. Recall that the mean value of a function f ∈ L 1 loc (R) on an interval I is defined as For an interval U , we say that a function f ∈ BMO(U ) if For further properties of the BMO space, see for example [MMY97,Appendix].
In the following, we will state the main properties of this section which show that the BMO properties fundamentally differ between k-Brjuno functions and the Wilton function. We first give the statement for the k-Brjuno functions.
Contrarily, for the Wilton function we have the following statement: On the other hand, we define for the following g := √ 5−1 2 and have: Theorem 2.3. For all α ∈ 1 2 , g , the function W α is a 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 [MMY97, Thm. 3.2] from the classical Brjuno function to k-Brjuno functions.
As a comparison to Figure 1 showing B 1,1 and W 1 in Figure 2, some numerical estimates of W α with different values of α are shown. The numerical estimates suggest that also for α ∈ (g, 1) the function W α is a BMO function. However, unfortunately, the results from Proposition 2.3 can not immediately be transferred to α ∈ (g, 1), see Remark 2.10 for an explanation which difficulties occur.
2.1. Proof that the real k-Brjuno functions are 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,α differs from B k,1 only by an L ∞ function which follows from Propositions 2.5 and 2.8. As the proof follows from very similar arguments as those in [MMY97,Thm. 3.2], we will only describe shortly the necessary changes in the proofs. (upper right), α = e − 2 (lower left) and α = 0.9 (lower right).
We start by introducing (2.1) endowed with a norm which is the sum of the BMO seminorm 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: . For all k ∈ N, the operator T k,1/2 as in (1.6) is a bounded linear operator from X * to X * whose spectral radius is bounded by To proceed, we prove an analog of [MMY97, Prop. 2.3, eq. (iv)].
Proposition 2.5. For all k ∈ N, there exists a constant C 1,k > 0 such that for all α ∈ [1/2, 1] and x ∈ R\Q, one has Before we start with the proof, we recall the following property.
Proof. For the following calculations we drop the dependence on α and x. We obtain by analogous calculations as in [MMY97, Prop. 2.3, eq. (iv)] that We notice that |β i | ≤ 1 and thus where the last estimate follows as in the proof of [MMY97, Prop. 2.3, eq. (iv)] and c 2 is given in Remark (2.6). Furthermore, which also follows as in the proof of [MMY97, Prop. 2.3, eq. (iv)] and c 1 is given in Remark (2.6). Finally, we have 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 α ∈ [1/2, 1] and x ∈ R\Q, one has The next proposition is an equivalent to [MMY97, Prop. 2.4].
Proposition 2.8. For all k ∈ N, there exists a constant C 2,k > 0 such that for all α ∈ [1/2, 1] and for all x ∈ R\Q one has Proof. The proof follows completely analogously to that of [ (which in this publication is denoted by Q j ) in the denominator is replaced by (q (1) j ) k . Proof of Proposition 2.1. By Proposition 2.4, 1−T k,α is invertible on X * . A Z-periodic even function equal to − log x on (0, 1/2] is in X * . Thus, B k,1/2 is BMO. Since by Proposition 2.8, the k-Brjuno function B k,α differs from B k,1/2 only by an L ∞ function, also B k,α for α ∈ [1/2, 1] is a BMO function.

Proofs of the BMO properties of the Wilton function.
Proof of Theorem 2.2. For brevity, in the following we write O I (f ) = 1 |I| I |f (x) − f I |dx. Furthermore, as we only consider α = 1, we also always write W instead of W 1 . By [MMY95, Proposition A.7], if I 1 and I 2 are two consecutive intervals, then (2.2) Let I n := − 1 n , 1 n = − 1 n , 0 ∪ 0, 1 n =: thus we clearly have W I + n = log(n) + 1 + O(1/n).
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 BMO function. Hence, comparing W α for α < 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 BMO function and compare W 1/2 with W α for α ∈ [1/2, g], see Proposition 2.9.
See Figure 3 for the graphs of A 1/2 and A α for a typical α ∈ (1/2, g).
where 3 in the continued fraction expansion appears i times. Note that r i = s i−1 and 2 − 1 1−t i = r i s i . More precisely, Then, A 1/2 (t i ) = t i−1 , A α ( r i s i ) = A 1/2 ( r i s i ) = r i−1 s i−1 for i ≥ 1 and t i 1 − g and r i s i 1 − g. Now, we suppose that n is the minimal index such that n , a (1/2) n = a (α) n + 1 ≥ 3, (1/2) n = −1, (α) n = 1 and q (1/2) n = q (α) n + q (1/2) n−1 . Since n and the domain of A 1/2 is [0, 1 2 ], we have On the other hand, x

Then, q
(1/2) n+i . In the following for we denote by g.z = az+b cz+d the Möbius transform applied on z. With this notation we have n+i .
Proof of Theorem 2.3. As we already observed 1 − S (1/2) is invertible in X * , which together with the fact that the Z-periodic even function equal to − log x on (0, 1/2] is in X * implies that W 1/2 is BMO. By Proposition 2.9, W α is BMO for α ∈ [1/2, g]. Finally, we give a remark which difficulties occur if one wants to extend the results of Proposition 2.3 to α ∈ (g, 1).
Remark 2.10. For the case α > g, we cannot directly apply the same argument as in the proof of Proposition 2.9. If α > g, then A α has a branch which is defined by 1/x − 1 (see Figure 4 for the graph of A α ) contrary to the case of α ≤ g. It causes a different behaviour of the orbits of the points under A α . In the proof, we showed a relation between x (1/2) n and x (α) n . By following the same argument for α > g, we only obtain a relation between x

Behaviour of the truncated real Brjuno function and the truncated real Wilton function
For x ∈ R, recall that β −1 = 1 and 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, contrary to the previous section, we omit the α in β (α) j as we will always assume α to be one. In this section, we are interested to compare a finite k-Brjuno sum or finite Wilton sum with the k-Brjuno or Wilton of its principal convergent. For doing 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 p j /q j = m 0 + 1 m 1 + 1 m 2 + . . . + 1 m r with m r ≥ 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 , . . . , m r , . . .].) With this we can define the truncated real Brjuno function by and the truncated real Wilton function by Before stating the results of this section, we also introduce the notation x j = G j (x) for x ∈ (0, 1). This enables us to state the next two lemmas which are analogs to [MMY01,Lemma 5.20].
By letting C k := 2kC , we complete the proof.