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

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

Therefore, (3.10) leads us to

$$\begin{aligned} \begin{aligned} \Vert D_{p,M_\eta ,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}^2&\ge 4\pi ^{-2}\sin (2\pi p^{-4})p^{-3}\sum _{\ell _{g}\in \mathcal {L}^0_{p,M_\eta ,\alpha }}p^{-2(\ell _g-2)}p^{2(\ell _g-2)}\\&=4\pi ^{-2}\sin (2\pi p^{-4})p^{-3}\#\mathcal {L}^0_{p,M_\eta ,\alpha }, \end{aligned} \end{aligned}$$

and, by (3.9), we have

$$\begin{aligned} \#\mathcal {L}^0_{p,M_\eta ,\alpha }=\#\mathcal {L}^0_{p,{p^{\eta -3}},\alpha }\ge 1+\#\mathcal {L}^0_{p,{p^{\eta }},\alpha }\ge 1+\frac{1}{4}\#\mathcal {L}_{p,p^\eta ,\alpha }. \end{aligned}$$

Hence, letting \(\eta \rightarrow \infty \), we obtain that our initial claim holds. \(\square \)