## 1 Introduction and statement of the result

Let $${\mathbb {N}}$$ denote the set of all non-negative integers, and let $${\mathbb {T}}=\mathbb {R/Z}$$ denote the one dimensional torus. For every $$x\in {\mathbb {R}}$$, let $$\{x\}$$ represent the fractional part of x and let $$\lfloor x\rfloor$$ represent the integer part of x. Also, set $$\Vert x\Vert =\min (\{x\},1-\{x\})$$ as the distance from x to $${\mathbb {Z}}$$.

A sequence $$\{x_n\}_{n\in {\mathbb {N}}}$$ is said to be uniformly distributed in $${\mathbb {T}}$$ if for any interval $$I\subseteq {\mathbb {T}}$$ with measure |I|, it holds

\begin{aligned} \lim _{N\rightarrow \infty }\frac{\#\{x_n\in I\mid 0\le n\le N-1\}}{N}=|I|. \end{aligned}

We introduce the concept of discrepancy as a quantitative counterpart of the uniform distribution. Namely, for $$N\in {\mathbb {N}}$$, we define the discrepancy of a sequence $$\{x_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {T}}$$ with respect to intervals as

\begin{aligned} D_N(\{x_n\}_{n\in {\mathbb {N}}}):=\sup _{I\subset {\mathbb {T}},\, I\text { interval}}\left| \sum _{n=0}^{N-1}\mathcal {X}_I(x_n)-N|I|\right| , \end{aligned}

where $$\mathcal {X}_{I}:{\mathbb {T}}\rightarrow \{0,1\}$$ is the indicator function of the interval $${I}\subset {\mathbb {T}}$$.

In 1935, J.G. van der Corput conjectured that for any sequence of real numbers $$\{x_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {T}}$$, the quantity $$D_N(\{x_n\}_{n\in {\mathbb {N}}})$$ is unbounded with respect to N. This conjecture was first proved ten years later by van Aardenne-Ehrenfest [14] with a first lower bound, and in 1954, Roth [10] significantly improved this estimate with one of the most relevant results in discrepancy theory. Later in 1972, Schmidt [12] gave the best possible estimate (up to a multiplicative constant). Namely, he showed that for any sequence $$\{x_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {T}}$$, it holds the lower bound

\begin{aligned} \limsup _{N\rightarrow \infty }\frac{D_N(\{x_n\}_{n\in {\mathbb {N}}})}{\log (N)}>10^{-2}. \end{aligned}

We formally introduce the object of our studies. Let $$\mathbb {N_+}$$ denote the set of all positive natural numbers, and consider $$p\ge 2$$. For $$n\in {\mathbb {N}}$$ with p-adic expansion

\begin{aligned} n=\sum _{j=1}^{\eta }n_jp^{j-1}\text { with }\eta \in \mathbb {N_+}\text { and }0\le n_j<p\text { for all } j, \end{aligned}

we define the p-adic van der Corput sequence $$\{x_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {T}}$$ as the sequence described by

\begin{aligned} x_n=\sum _{j=1}^{\eta }n_jp^{-j}. \end{aligned}

In 1935, van der Corput [15] showed that the discrepancy of the dyadic van der Corput sequence is optimal with respect to the order of magnitude in N, and it is well-known that the same holds for all p-adic van der Corput sequences. In particular, Faure [4] generalized p-adic van der Corput sequences and further investigated their qualities. Since its low-discrepancy property, the distribution of the van der Corput sequence has been studied under various points of view. We refer the reader to [6] for a detailed survey on all the generalizations of such sequence and the concerning properties.

We introduce a major tool in discrepancy theory. Consider a fixed anchored interval $$[0,\alpha )\subset {\mathbb {T}}$$ and, for $$N\in \mathbb {N_+}$$, define the discrepancy function of the p-adic van der Corput sequence $$\{x_n\}_{n\in {\mathbb {N}}}$$ with respect to $$[0,\alpha )$$ as

\begin{aligned} D_{p,N,\alpha }:=\sum _{n=0}^{N-1}\mathcal {X}_{[0,\alpha )}(x_n)-N\alpha . \end{aligned}

In 1972, Schmidt [11] first showed that if $$\alpha$$ has finite dyadic expansion, then $$D_{2,N,\alpha }$$ is bounded, and in 1978, Shapiro [13] proved the converse implication through ergodic theory methods. Later in 1980, Hellekalek [7] generalized the results for p-adic van der Corput sequences.

In 1983, Faure [5] gave an explicit formula for $$D_{p,N,\alpha }$$. For completeness, we report his result as follows. Let $$N\in \mathbb {N_+}$$, and write the p-adic expansions $$\alpha =\sum _{j=0}^{\infty }\alpha _jp^{-j-1}$$ and $$N=\sum _{i=0}^{\infty }N_ip_i$$. Now, set $$V(N,\alpha )=\sum _{i=0}^{\infty }\inf (N_i,\alpha _i)$$ and denote $$W(N,\alpha )$$ as the number of couples $$(-,+)$$ in the sequences of the signs of $$\{\alpha _i-N_i\}_{i\ge 0}$$ without considering the zeros. Then it holds

\begin{aligned} D_{p,N,\alpha }=-\sum _{j=0}^{\infty }\sum _{i=0}^{j}N_i\alpha _jp^{i-j-1}+V(N,\alpha )+W(N,\alpha ). \end{aligned}

Further useful estimates for $$D_{2,N,\alpha }$$ can be found in [3].

It is now time to introduce the main tool of this paper.

### Definition 1.1

Consider a fixed interval $$[0,\alpha )\subset {\mathbb {T}}$$. For $$N\in \mathbb {N_+}$$, we define the geometrically-shifted discrepancy function of the p-adic van der Corput sequence $$\{x_n\}_{n\in {\mathbb {N}}}$$ with respect to $$[0,\alpha )$$ as

\begin{aligned} D_{p,N,\alpha }(t):=\sum _{n=0}^{N-1}\mathcal {X}_{[0,\alpha )}(x_n+t)-N\alpha . \end{aligned}
(1.1)

One can interpret (1.1) as the usual discrepancy function of the p-adic van der Corput sequence with respect to a shifted interval $$[-t,-t+\alpha )\subset {\mathbb {T}}$$. In particular, $$D_{p,N,\alpha }(0)=D_{p,N,\alpha }$$.

The aim of this paper is to investigate the asymptotic behavior of $$\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}$$, i.e., the root mean square average of the discrepancy function over all possible intervals of length $$\alpha$$ in $${\mathbb {T}}$$.

We remark that the quantity

\begin{aligned} \left( \mathop {\int }\limits _{0}^{1}\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2\,d\alpha \right) ^{\frac{1}{2}} \end{aligned}

is exactly the periodic $$L^2$$-discrepancy introduced by Lev [9] in 1995. Moreover, we directly deduce from Lemma 2.2 that the latter quantity coincides with the diaphony introduced by Zinterhof [16] in 1976 (see [1, Proposition 2] for a generalization of such equality in higher dimensions). As a matter of fact, the periodic $$L^2$$-discrepancy is a geometric interpretation of diaphony, and it is also closely related to the worst-case integration error of quasi-Monte Carlo integration rules (see, for example, [8]).

We need a last definition in order to state our main result.

### Definition 1.2

For $$\alpha$$ with p-adic expansion $$\alpha =\sum _{\ell =1}^{\infty }\alpha _\ell p^{-\ell }$$ and for N with p-adic expansion $$N=\sum _{j=1}^{\eta }N_jp^{j-1}$$ (with $$N_\eta \ne 0$$), we define

\begin{aligned} \mathcal {L}_{p,N,\alpha }:=\{1\le \ell \le \lfloor \log _p(N)\rfloor \text { such that }\alpha _\ell \not \in \{0,p-1\}\text { or }\alpha _{\ell -1}\ne \alpha _{\ell }\}. \end{aligned}

In other words, $$\mathcal {L}_{p,N,\alpha }$$ represents the set of the indices $$\ell \in \left[ 1,\eta -1\right]$$ such that the coefficient $$\alpha _\ell$$ does not lie on the extremities $$\{0,p-1\}$$ or it belongs to $$\{0,p-1\}$$ but it is different from the previous coefficient. We further refer to Faure’s [5, Theorem 1] on generalized van der Corput sequences for similarities between the quantities involved.

Then, in relation to the p-adic expansion of $$\alpha$$, we can give an asymptotic characterization of $$\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}$$ with respect to N. We state the main result of this paper.

### Theorem 1.3

For $$\alpha \in (0,1)$$ and $$p\in \mathbb {N_+}$$, we have

\begin{aligned} 0<\limsup _{N\rightarrow \infty }\frac{\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2}{\#\mathcal {L}_{p,N,\alpha }}<\infty . \end{aligned}

It directly follows a characterization of the intervals for which the $$L^2$$-norm of the geometrically-shifted discrepancy function is bounded.

### Corollary 1.4

Consider $$\alpha \in (0,1)$$ and $$p\in \mathbb {N_+}$$, then $$\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}$$ is bounded with respect to N if and only if $$\alpha$$ has finite p-adic expansion.

## 2 Auxiliary results for the proof

From now on, let $$\{x_n\}_{n\in {\mathbb {N}}}$$ denote the p-adic van der Corput sequence. Moreover, improper sums (as $$\sum _{i=1}^{0}1$$) will be conventionally considered as zeros.

Notice that for any $$\eta \in {\mathbb {N}}$$, we have

\begin{aligned} \{x_n\}_{n=0}^{p^\eta -1}=\Big \{\sum _{j=1}^{\eta }n_jp^{-j}\text { such that }0\le n_j<p\text { for all } j\Big \}=\left\{ \frac{k}{p^{\eta }}\right\} _{k=0}^{p^\eta -1}. \end{aligned}

More precisely, for any $$\eta \in {\mathbb {N}}$$, the first $$p^\eta$$ points in the p-adic van der Corput sequence are $$p^\eta$$ equidistant points in $${\mathbb {T}}$$ and, trivially, the point 0 is the first of them. We extend this fact in the following lemma.

### Lemma 2.1

Given $$h\in \mathbb {N_+}$$ and $$N\in \mathbb {N_+}$$ such that N has p-adic expansion

\begin{aligned} N=\sum _{j=h}^{\eta }N_jp^{j-1}\text { with }0\le N_j<p\text { for all } j, \end{aligned}

then $$\{x_n\}_{n=N}^{N+p^{h-1}-1}$$ are $$p^{h-1}$$ equidistant points in $${\mathbb {T}}$$, moreover the point $$\sum _{j=h}^{\eta }N_jp^{-j}$$ is one of them.

### Proof

We can uniquely write all the numbers k such that $$N\le k\le N+p^{h-1}-1$$ as p-adic expansions

\begin{aligned} k=\sum _{j=h}^{\eta } N_jp^{j-1}+\sum _{j=1}^{h-1}\beta _jp^{j-1}\text { where }0\le \beta _j<p\text { for all } j. \end{aligned}

This consequently means that all the elements $$x_k$$ in $$\{x_n\}_{n=N}^{N+p^{h-1}-1}$$ can uniquely be written as

\begin{aligned} x_k=\sum _{j=h}^{\eta }N_jp^{-j}+\sum _{j=1}^{h-1}\beta _jp^{-j}\text { where }0\le \beta _j<p\text { for all } j. \end{aligned}

The first sum is then a constant term for all these $$x_k$$, while the second sum is the same one that appears for the first $$p^{h-1}$$ elements of the van der Corput sequence. Therefore $$\{x_n\}_{n=N}^{N+p^{h-1}-1}$$ are $$p^{h-1}$$ equidistant points of $${\mathbb {T}}$$, and $$\sum _{j=0}^{\eta }N_jp^{-j}$$ is one of them. $$\square$$

Now, we need to recall some fundamental properties of Fourier analysis. For $$f\in L^2({\mathbb {T}})$$, we define the Fourier transform of f as the function $$\mathcal {F}f:{\mathbb {Z}}\rightarrow {\mathbb {C}}$$ described by

\begin{aligned} (\mathcal {F}f)(k)=\mathop {\int }\limits _{\mathbb {T}}f(x)\exp (-2\pi ixk)\,dx\qquad \forall k\in {\mathbb {Z}}. \end{aligned}

For $$y\in {\mathbb {R}}$$, we use the notations $$\tau _yf=f(\cdot -y)$$ and $$e_y(k)=\exp (2\pi iy k)$$. It is then a well-known property of Fourier transforms that $$\mathcal {F}(\tau _yf)=e_{-y}\,\mathcal {F}f$$. Therefore, considering indicator functions of intervals, we get

\begin{aligned} \mathcal {F}(\mathcal {X}_{[0,\alpha )}(x_n+\cdot ))=\mathcal {F}(\tau _{-x_n}\mathcal {X}_{[0,\alpha )})=e_{x_n}\,\mathcal {F}\mathcal {X}_{[0,\alpha )}. \end{aligned}
(2.1)

We show a fundamental lemma that works as starting point for our results.

### Lemma 2.2

Consider an interval $${[0,\alpha )}\subset {\mathbb {T}}$$, then

\begin{aligned} \Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2=\sum _{k\in {\mathbb {Z}}\setminus \{0\}}|\pi k|^{-2}|\sin (\pi \alpha k)|^2\left| \sum _{n=0}^{N-1}\exp (2\pi i kx_n)\right| ^2. \end{aligned}

### Proof

It is obvious that $$D_{p,N,\alpha }(\cdot )$$ is in $$L^2({\mathbb {T}})$$, so that we can apply Parseval’s identity and, by (2.1), we get

\begin{aligned} \begin{aligned} \Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2&=\sum _{k\in {\mathbb {Z}}}|\mathcal {F}D_{p,N,\alpha }(k)|^2\\&=\sum _{k\in {\mathbb {Z}}\setminus \{0\}}\left| \sum _{n=0}^{N-1}\exp (2\pi i kx_n)\right| ^2\left| \mathcal {F}\mathcal {X}_{[0,\alpha )}(k)\right| ^2 \end{aligned} \end{aligned}

since it easily follows that $$\mathcal {F}D_{p,N,\alpha }(0)=0$$. Moreover, we have

\begin{aligned} \begin{aligned} \mathcal {F}\mathcal {X}_{[0,\alpha )}(k)&=\mathop {\int }\limits _{{\mathbb {T}}}\mathcal {X}_{[0,\alpha )}(x)\exp (-2\pi i kx)dx=(-2\pi ik)^{-1}(\exp (-2\pi i k\alpha )-1)\\&=(2\pi ik)^{-1}\exp (-\alpha \pi ik)\left[ \exp (+\alpha \pi ik)-\exp (-\alpha \pi ik)\right] \\&=(\pi k)^{-1}\exp (-\alpha \pi ik)\left[ \sin (\pi \alpha k)\right] , \end{aligned} \end{aligned}

so that we get our initial claim. $$\square$$

## 3 The proof of the main result

The results in this section are inspired by [2], and we follow its notation. For any positive integer q, we define the function $$\delta _q:{\mathbb {Z}}\rightarrow \{0,1\}$$ as

\begin{aligned} \delta _q(h)={\left\{ \begin{array}{ll} 1&{} \text {if }h\equiv 0\ (\textrm{mod}\ q),\\ 0&{} \text {if }h\not \equiv 0\ (\textrm{mod}\ q). \end{array}\right. } \end{aligned}

Moreover, the sum of a geometric series gives

\begin{aligned} \delta _q(h)=\frac{1}{q}\sum _{s=0}^{q-1}\exp \left( 2\pi i h \frac{s}{q}\right) . \end{aligned}
(3.1)

Now, consider $$N\in {\mathbb {N}}$$ with p-adic expansion $$N=\sum _{j=1}^{\eta }N_jp^{j-1}$$ and recall that, by Lemma 2.2, we have

\begin{aligned} \Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2=\sum _{k\in {\mathbb {Z}}\setminus \{0\}}|\pi k|^{-2}|\sin (\pi \alpha k)|^2\left| \sum _{n=0}^{N-1}\exp (2\pi i kx_n)\right| ^2. \end{aligned}
(3.2)

Because of Lemma 2.1, we are able to split $$\{x_n\}_{n=0}^{N-1}$$ in disjoint sets of equidistant points. More precisely, we get

\begin{aligned} \begin{aligned}&\{x_n\}_{n=0}^{N-1}=\\&=\bigsqcup _{\nu =0}^{\eta -1}\bigsqcup _{d=0}^{N_{\eta -\nu }-1}\!\Big \{sp^{-\eta +\nu +1}+dp^{-\eta +\nu }+\!\sum _{\mu =0}^{\nu -1}N_{\eta -\mu }p^{-\eta +\mu }\mid 0\le s\le p^{\eta -\nu -1}-1 \Big \}, \end{aligned} \end{aligned}

where as the parameter $$\nu$$ increases, we consider (exponentially) smaller sets of equidistant points in $${\mathbb {T}}$$, while the parameter d is less significant but nonetheless necessary.

Hence, we can also split the sum in the previous modulus in several parts, namely

\begin{aligned} \begin{aligned}&\sum _{n=0}^{N-1}\exp (2\pi i k x_n)=\\&=\sum _{\nu =0}^{\eta -1}\sum _{d=0}^{N_{\eta -\nu }-1}\sum _{s=0}^{p^{\eta -\nu -1}-1}\exp \Big (2\pi i k\Big (sp^{-\eta +\nu +1}+dp^{-\eta +\nu }+\sum _{\mu =0}^{\nu -1}N_{\eta -\mu }p^{-\eta +\mu }\Big )\Big ). \end{aligned} \end{aligned}

Although the long expression, we are just reordering and writing explicitly all these sets because each one of them represents a regular polygon lying on the unit circle in the complex plane. Hence, from (3.1), we further get

\begin{aligned} \begin{aligned}&\!\!\sum _{n=0}^{N-1}\exp (2\pi i k x_n)=\\&\!\!=\sum _{\nu =0}^{\eta -1}p^{\eta -\nu -1}\delta _{p^{\eta -\nu -1}}\!(k)\!\sum _{d=0}^{N_{\eta -\nu }-1}\!\!\exp \Big (2\pi i k\Big (dp^{-\eta +\nu }+\!\sum _{\mu =0}^{\nu -1}\!N_{\eta -\mu }p^{-\eta +\mu }\Big )\Big ). \end{aligned}\nonumber \\ \end{aligned}
(3.3)

Before giving the proof of our main result, let us first show a simpler one.

### Proposition 3.1

Let $$\lambda \in \mathbb {N_+}$$ and let $${[0,\alpha )}$$ be such that $$\alpha =\sum _{\ell =1}^{\lambda }\alpha _\ell p^{-\ell }$$ with $$0\le \alpha _\ell <p$$ for all $$\ell$$ (i.e., $$\alpha$$ has finite p-adic expansion), then $$\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}$$ is bounded with respect to N by a constant depending only on p and $$\lambda$$.

### Proof

It is easy to see that

\begin{aligned} |\sin (\pi \alpha k)|\le 1-\delta _{p^\lambda }(k). \end{aligned}
(3.4)

Moreover, notice that for $$a,b\in \mathbb {N_+}$$ such that $$a\le b$$, we have

\begin{aligned} (1-\delta _{p^a}(k))\,\delta _{p^b}(k)=0\quad \forall k\in {\mathbb {Z}}. \end{aligned}
(3.5)

Consider $$\eta \in \mathbb {N_+}$$ such that $$\eta \ge \lambda$$, and let $$\sum _{j=1}^{\eta }N_jp^{j-1}$$ be the p-adic expansion of N. Starting with (3.2) and by (3.3)-(3.4)-(3.5), it follows that

\begin{aligned} \begin{aligned} \Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2&\le \pi ^{-2}\sum _{k\in {\mathbb {Z}}\setminus \{0\}}|k|^{-2}(1-\delta _{p^\lambda }(k))^2\,\left| \sum _{\nu =0}^{\eta -1}p^{\eta -\nu -1}\delta _{p^{\eta -\nu -1}}(k)\,\cdot \right. \\&\left. \qquad \cdot \sum _{d=0}^{N_{\eta -\nu }-1}\exp \Big (2\pi i k\Big (dp^{-\eta +\nu }+\sum _{\mu =0}^{\nu -1}N_{\eta -\mu }p^{-\eta +\mu }\Big )\Big )\right| ^2\\&\le \pi ^{-2}\sum _{k\in {\mathbb {Z}}\setminus \{0\}}|k|^{-2}\left| \sum _{\nu =\eta -\lambda }^{\eta -1}p^{\eta -\nu -1}\delta _{p^{\eta -\nu -1}}(k)\,\cdot \right. \\&\left. \qquad \cdot \sum _{d=0}^{N_{\eta -\nu }-1}\exp \Big (2\pi i k\Big (dp^{-\eta +\nu }+\sum _{\mu =0}^{\nu -1}N_{\eta -\mu }p^{-\eta +\mu }\Big )\Big )\right| ^2. \end{aligned} \end{aligned}

With the change of variable $$\tau =\eta -\nu -1$$, and since $$0\le N_j<p$$ for all j, we finally get

\begin{aligned} \Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2\le \pi ^{-2}\sum _{k\in {\mathbb {Z}}\setminus \{0\}}|k|^{-2}\left| (p-1)\sum _{\tau =0}^{\lambda -1}p^{\tau }\delta _{p^{\tau }}(k)\right| ^2, \end{aligned}

which is clearly bounded by a constant depending only on p and $$\lambda$$. $$\square$$

It is time to prove the main result of this paper.

### Proof of Theorem 1.3

We break the proof in two parts, first we show that the right inequality holds.

As before, let $$\sum _{j=1}^{\eta }N_jp^{j-1}$$ (with $$N_\eta \ne 0$$) be the p-adic expansion of N. Let k be a positive integer, we respectively define L(k) and G(k) as the lowest and the greatest exponents (plus 1) appearing in the p-adic expansion of k, namely $$k=\sum _{m=L(k)}^{G(k)}\beta _mp^{m-1}$$ with $$\beta _{L(k)}\ne 0$$ and $$\beta _{G(k)}\ne 0$$.

In the case of $$L(k)\le \eta$$, starting with (3.3) and recalling that $$0\le N_j<p$$ for all j, we get

\begin{aligned} \begin{aligned} \left| \sum _{n=0}^{N-1}\exp (2\pi ikx_n)\right|&\le (p-1)\left( \sum _{\nu =0}^{\eta -1}p^{\eta -\nu -1}\delta _{p^{\eta -\nu -1}}(k)\right) \\&=(p-1)\left( \sum _{\nu =\eta -L(k)}^{\eta -1}p^{\eta -\nu -1}\right) \\&=(p-1)\left( \frac{p^{L(k)}-1}{p-1}\right) =p^{L(k)}-1 \end{aligned} \end{aligned}

since k is a multiple of $$p^{\eta -\nu -1}$$ if and only if $$L(k)-1\ge \eta -\nu -1$$. Trivially it holds $$N\le p^\eta$$, so that for any choice of $$k\in {\mathbb {Z}}$$, we get

\begin{aligned} \left| \sum _{n=0}^{N-1}\exp (2\pi ikx_n)\right| \le p^{\min (L(k),\eta )}. \end{aligned}

Also, it is a well-known inequality that

\begin{aligned} \pi \Vert x\Vert \ge |\sin (\pi x)|\ge 2\Vert x\Vert \qquad \forall x\in {\mathbb {R}}, \end{aligned}
(3.6)

therefore, starting with (3.2), we get

\begin{aligned} \Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2\le \sum _{k\in {\mathbb {Z}}\setminus \{0\}}|k|^{-2}\Vert \alpha k\Vert ^2 p^{2\min (L(k),\eta )}. \end{aligned}
(3.7)

Now, write $$\mathcal {L}_{p,N,\alpha }$$ as an increasing sequence of indices $$\{\ell _q\}_{q=1}^{\#\mathcal {L}_{p,N,\alpha }}$$ and, for the sake of construction, add $$\ell _0=0$$ and $$\ell _{{1+\#\mathcal {L}_{p,N,\alpha }}}=\eta$$. Notice that all the indices $$\ell$$ (if any) in between two consecutive $$\ell _q$$ are such that the respective coefficients $$\alpha _\ell$$ are either all 0 or all $$p-1$$.

In the case of $$\ell _q< L(k)\le G(k)\le \ell _{q+1}$$, we proceed to show that

\begin{aligned} \Vert \alpha k\Vert \le p^{G(k)-\ell _{q+1}+1}. \end{aligned}

In fact, suppose without loss of generality that all the $$\ell$$ such that $$\ell _q<\ell <\ell _{q+1}$$ are zeros (the other case is similar), so that we get

\begin{aligned} \begin{aligned} \Vert \alpha k\Vert&=\Vert \sum _{\ell =1}^{\infty }\alpha _\ell p^{-\ell }\cdot \sum _{m=L(k)}^{G(k)}\beta _mp^{m-1}\Vert \\&=\Vert (\sum _{\ell =1}^{\ell _q}+\sum _{\ell =\ell _{q+1}}^{\infty })\alpha _\ell p^{-\ell }\cdot \sum _{m=L(k)}^{G(k)}\beta _mp^{m-1}\Vert \\&=\Vert \sum _{\ell =\ell _{q+1}}^{\infty }\alpha _\ell p^{-\ell }\cdot \sum _{m=L(k)}^{G(k)}\beta _mp^{m-1}\Vert \le \Big \{\sum _{\ell =\ell _{q+1}}^{\infty }\alpha _\ell p^{-\ell }\cdot \sum _{m=L(k)}^{G(k)}\beta _mp^{m-1}\Big \}\\&=\sum _{\ell =\ell _{q+1}}^{\infty }\alpha _\ell p^{-\ell }\cdot \sum _{m=L(k)}^{G(k)}\beta _mp^{m-1}\le p^{-\ell _{q+1}+1}p^{G(k)}= p^{G(k)-\ell _{q+1}+1}. \end{aligned} \end{aligned}

In the case $$L(k)\le \ell _q<G(k)\le \ell _{q+1}$$, we trivially get $$\Vert \alpha k\Vert \le 1$$. Then it is useful to define

\begin{aligned} f_q(r,s)={\left\{ \begin{array}{ll} p^{s-\ell _{q+1}+1}&{}\text {if}\quad \ell _q< r\le s\le \ell _{q+1},\\ 1&{}\text {else}. \end{array}\right. } \end{aligned}

Moreover, notice that $$\#\{k\in {\mathbb {N}}\mid L(k)=r, G(k)=s\}\le p^{s-r+1}$$ and that trivially $$p^{G(k)-1}\le |k|< p^{G(k)}$$.

Finally, taking into account what we have gathered so far and reordering the sum in (3.7), we get

\begin{aligned} \begin{aligned}&\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2\le \sum _{k\in {\mathbb {Z}}\setminus \{0\}}|k|^{-2}\Vert \alpha k\Vert ^2 p^{2\min (L(k),\eta )}\le \\&\le 2\Big (\sum _{q=0}^{\#\mathcal {L}_{p,N,\alpha }}\!\sum _{G=\ell _{q}+1}^{\ell _{q+1}}+\sum _{G=\eta +1}^{\infty }\Big )\sum _{L=1}^{G}p^{G-L+1}p^{-2G+2}f_q(L,G)^2\,p^{2\min (L,\eta )}, \end{aligned} \end{aligned}

so, writing explicitly all the sums, it follows that

\begin{aligned} \frac{1}{2}\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2&\le \sum _{q=0}^{\#\mathcal {L}_{p,N,\alpha }}\sum _{G=\ell _{q}+1}^{\ell _{q+1}}\Big (\sum _{L=1}^{\ell _q}p^{G-L+1}p^{-2G+2}p^{2L}\\&\quad +\sum _{L=\ell _{q}+1}^{G}p^{G-L+1}p^{-2G+2}p^{2G-2\ell _{q+1}+2}p^{2L}\Big )\\&\quad +\sum _{G=\eta +1}^{\infty }\sum _{L=1}^{G}p^{G-L+1}p^{-2G+2}p^{2\min (L,\eta )}\\&\le \sum _{q=0}^{\#\mathcal {L}_{p,N,\alpha }}\sum _{G=\ell _{q}+1}^{\ell _{q+1}}\Big (p^3\sum _{L=1}^{\ell _q}p^{-G+L}+p^5\sum _{L=\ell _{q}+1}^{G}p^{G+L-2\ell _{q+1}}\Big )\\&\quad +p^3\sum _{G=\eta +1}^{\infty }\Big (\sum _{L=1}^{\eta }p^{-G+L}+\sum _{L=\eta +1}^{G}p^{-G-L+2\eta }\Big )\\&\le 2p^5\sum _{q=0}^{\#\mathcal {L}_{p,N,\alpha }}\sum _{G=\ell _{q}+1}^{\ell _{q+1}}\left( p^{-G+\ell _q}+p^{2G-2\ell _{q+1}}\right) \\&\quad +2p^3\sum _{G=\eta +1}^{\infty }\left( p^{-G+\eta }+p^{-G+\eta }\right) \\&\le 4p^3+2p^5\sum _{q=0}^{\#\mathcal {L}_{p,N,\alpha }}3\le 7p^5\#\mathcal {L}_{p,N,\alpha }, \end{aligned}

so that our initial claim on the right inequality holds.

Now, we proceed to show that the left inequality holds. Notice that for $$\ell _q\in \mathcal {L}_{p,N,\alpha }$$ it holds

\begin{aligned} \Vert p^{\ell _q-2}\alpha \Vert =\Vert p^{\ell _q-2}\sum _{\ell =1}^{\infty }\alpha _\ell p^{-\ell }\Vert =\Vert p^{\ell _q-2}\sum _{\ell =\ell _q-1}^{\infty }\alpha _\ell p^{-\ell }\Vert =\Vert \sum _{n=1}^{\infty }\alpha _{\ell _q+n-2}p^{-n}\Vert , \end{aligned}

and, from Definition 1.2, it follows that

\begin{aligned} \sum _{n=1}^{\infty }\alpha _{\ell _q+n-2}p^{-n}\in \left( \frac{1}{p^2},\frac{p^2-1}{p^2}\right) , \end{aligned}

because the couple $$(\alpha _{\ell _q-1},\alpha _{\ell _q})\not \in \{(0,0),(p-1,p-1)\}$$. Therefore

\begin{aligned} \Vert p^{\ell _q-2}\alpha \Vert \ge p^{-2}. \end{aligned}
(3.8)

Now, for $$\theta \in \{0,1,2,3\}$$ and for $$N\in {\mathbb {N}}$$, consider

\begin{aligned} \mathcal {L}^\theta _{p,N,\alpha }:=\{\ell \text { such that }\ell \in \mathcal {L}_{p,N,\alpha }\text { and }\ell \equiv \theta \ (\textrm{mod}\ 4)\}, \end{aligned}

then for each $$\eta \in {\mathbb {N}}$$, there is a $$\theta _\eta \in \{0,1,2,3\}$$ such that

\begin{aligned} \#\mathcal {L}^{\theta _{\eta }}_{p,p^\eta ,\alpha }\ge \frac{1}{4}\,\#\mathcal {L}_{p,p^\eta ,\alpha } \end{aligned}
(3.9)

and, without loss of generality, we assume $$\theta _\eta \equiv 0$$ for all $$\eta$$.

Also, for any $$\eta \in {\mathbb {N}}$$ such that $$\eta \equiv 0\ (\textrm{mod}\ 4)$$, define

\begin{aligned} M_\eta =\sum _{m=1}^{\frac{\eta }{4}}p^{4m-3}=\sum _{n=1}^{\eta }\Delta _n\,p^{n-1}\quad \text {where }\,\Delta _n=\delta _{4}(n-2), \end{aligned}

then notice that $$p^{\eta -3}\le M_\eta < p^{\eta -2}$$, and therefore $$\#\mathcal {L}^0_{p,M_\eta ,\alpha }=\#\mathcal {L}^0_{p,p^{\eta -3},\alpha }$$.

Now, write $$\mathcal {L}^0_{p,M_\eta ,\alpha }$$ as an increasing sequence of indices $$\{\ell _g\}_{g=1}^{\#\mathcal {L}^0_{p,M_\eta ,\alpha }}$$ and finally, starting with (3.2) and by (3.6)–(3.8), we get

\begin{aligned} \begin{aligned}&\Vert D_{p,M_\eta ,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2\ge \\&\ge 4(\pi p)^{-2}\sum _{\ell _{g}\in \mathcal {L}^0_{p,M_\eta ,\alpha }}p^{-2(\ell _g-2)}\left| \sum _{n=0}^{M_\eta -1}\exp (2\pi i p^{\ell _g-2}x_n)\right| ^2. \end{aligned} \end{aligned}
(3.10)

Notice, by construction, that for $$\nu$$ and $$\ell _{g}$$ such that $$0\le \eta -\nu -1\le \ell _{g}-2$$ (or equivalently $$\delta _{p^{\eta -\nu -1}}(p^{\ell _{g}-2})=1$$), we have that the fractional part

\begin{aligned} \Big \{ p^{\ell _g-2}\sum _{\mu =0}^{\nu -1}\Delta _{\eta -\mu }p^{-\eta +\mu }\Big \}\in \left( \frac{1}{p^4},\frac{1}{p^3}\right) . \end{aligned}
(3.11)

Finally, starting with (3.3), we get

\begin{aligned} \begin{aligned}&\sum _{n=0}^{M_\eta -1}\exp (2\pi i p^{\ell _g-2} x_n)=\\&=\!\sum _{\nu =0}^{\eta -1}p^{\eta -\nu -1}\delta _{p^{\eta -\nu -1}}\!(p^{\ell _g-2})\!\!\sum _{d=0}^{\Delta _{\eta -\nu }-1}\!\!\exp \Big (2\pi i p^{\ell _g-2}\Big (dp^{-\eta +\nu }+\!\sum _{\mu =0}^{\nu -1}\!\Delta _{\eta -\mu }p^{-\eta +\mu }\Big )\!\Big )\\&=\sum _{\nu =0}^{\eta -1}p^{\eta -\nu -1}\delta _{p^{\eta -\nu -1}}(p^{\ell _g-2})\,\Delta _{\eta -\nu }\exp \Big (2\pi i p^{\ell _g-2}\sum _{\mu =0}^{\nu -1}\Delta _{\eta -\mu }\,p^{-\eta +\mu }\Big ), \end{aligned} \end{aligned}

so that, taking the imaginary part, we get

\begin{aligned} \begin{aligned}&\left| \sum _{n=0}^{M_\eta -1}\exp (2\pi i p^{\ell _g-2} x_n)\right| \ge \\&\ge \left| \sum _{\nu =0}^{\eta -1}p^{\eta -\nu -1}\delta _{p^{\eta -\nu -1}}(p^{\ell _g-2})\,\Delta _{\eta -\nu }\sin \Big (2\pi p^{\ell _g-2}\sum _{\mu =0}^{\nu -1}\Delta _{\eta -\mu }p^{-\eta +\mu }\Big )\right| . \end{aligned} \end{aligned}
(3.12)

Recall that the sine function $$\sin (2\pi \cdot )$$ is increasing and positive in the interval $$(0,\frac{1}{4})$$ so, from (3.11), it follows that

\begin{aligned} \sin \Big (2\pi p^{\ell _g-2}\sum _{\mu =0}^{\nu -1}\Delta _{\eta -\mu }p^{-\eta +\mu }\Big )\ge \sin (2\pi p^{-4}) \end{aligned}

whenever $$\eta -\nu -1\le \ell _g-2$$. On the other hand, whenever $$\eta -\nu -1>\ell _g-2$$, we trivially get $$\delta _{p^{\eta -\nu -1}}(p^{\ell _g-2})=0$$.

In conclusion, from (3.12), we get

\begin{aligned} \begin{aligned} \left| \sum _{n=0}^{M_\eta -1}\exp (2\pi i p^{\ell _g-2} x_n)\right|&\ge \sin (2\pi p^{-4})\left| \sum _{\nu =\eta -\ell _{g}+1}^{\eta -1}p^{\eta -\nu -1}\Delta _{\eta -\nu }\right| \\&\ge \sin (2\pi p^{-4})\sum _{\tau =0}^{\ell _{g}-2}p^{\tau }\Delta _{\tau +1}\ge \sin (2\pi p^{-4})p^{-1}p^{\ell _g-2}. \end{aligned} \end{aligned}

\begin{aligned} \begin{aligned} \Vert D_{p,M_\eta ,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2&\ge 4\pi ^{-2}\sin (2\pi p^{-4})p^{-3}\sum _{\ell _{g}\in \mathcal {L}^0_{p,M_\eta ,\alpha }}p^{-2(\ell _g-2)}p^{2(\ell _g-2)}\\&=4\pi ^{-2}\sin (2\pi p^{-4})p^{-3}\#\mathcal {L}^0_{p,M_\eta ,\alpha }, \end{aligned} \end{aligned}
\begin{aligned} \#\mathcal {L}^0_{p,M_\eta ,\alpha }=\#\mathcal {L}^0_{p,{p^{\eta -3}},\alpha }\ge 1+\#\mathcal {L}^0_{p,{p^{\eta }},\alpha }\ge 1+\frac{1}{4}\#\mathcal {L}_{p,p^\eta ,\alpha }. \end{aligned}
Hence, letting $$\eta \rightarrow \infty$$, we obtain that our initial claim holds. $$\square$$