A LeVeque-Type Inequality on the ring of $p$-adic integers

TL;DR

This paper establishes a LeVeque-type inequality for sequences in the ring of p-adic integers, providing bounds on their discrepancy using Fourier analysis, extending classical results to a p-adic setting.

Contribution

It introduces a p-adic analogue of the LeVeque inequality, applying Fourier analysis to derive discrepancy bounds on sequences in Zp.

Findings

Derived a discrepancy inequality for p-adic sequences

Applied the inequality to linear sequences in Zp

Extended classical discrepancy results to p-adic integers

Abstract

We derive an inequality on the discrepancy of sequences on the ring of $p$-adic integers $\ZZ_p$ using techniques from Fourier analysis. The inequality is used to obtain an upper bound on the discrepancy of the sequence $\alpha_n = na +b$, where $a$ and $b$ are elements of $\ZZ_p$. This is a $p$-adic analogue of the classical LeVeque inequality on the circle group $\RR/\ZZ$.

Equations

$${\mathbb{Z}}_p=\{x||x{|}_p\le 1\},$$

$$D(a,1/p^k)$$

$${\mathbb{Z}}_p=\bigcup _{j=0}^{p^k-1}D(j,1/p^k).$$

$$\underset{N\to \infty }{\lim }\left|\frac{\left|D(a,1/p^k)\cap \{{\alpha }_1,...,{\alpha }_N\}\right|}{N}-\frac{1}{p^k}\right|=0.$$

$$D_N={\sup }_{a\in {\mathbb{Z}}_p,k\in \mathbb{N}}\left|\frac{\left|D(a,1/p^k)\cap \{{\alpha }_1,...,{\alpha }_N\}\right|}{N}-\frac{1}{p^k}\right|.$$

$$\frac{1}{N}\le D_N\le 1.$$

$${\zeta }^x={\zeta }^{a_0+a_{1}p+a_2{p}^2+....+a_{n-1}{p}^{n-1}}.$$

$$D_N$$

$$D_N=O\left(N^{-1/2}\right).$$

$$\Psi :\mathbb{Z}(p^{\infty })$$

$$\zeta$$

$$\gamma (x)={\zeta }^x,$$

$$\hat{f}(\zeta )={\int }_{{\mathbb{Z}}^p}f(x){\zeta }^{-x}d\mu ,$$

$$f(x)=\sum _{\zeta \in \mathbb{Z}(p^{\infty })}\hat{f}(\zeta ){\zeta }^x,$$

$$\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{n=1}^N{\zeta }^{{\alpha }_n}=0.$$

$${\int }_{{\mathbb{Z}}_p}{\mathcal{X}}_{D(a,1/p^k)}(x)f(x)d\mu (x)=\frac{1}{p^k}{\int }_{{\mathbb{Z}}_p}f(a+p^{k}x)d\mu (x).$$

$$\widehat{{\mathcal{X}}}_{D(a,1/p^{k})}(\zeta)=\left\{\begin{array}[]{cc}\zeta^{-a}p^{-k}&\text{if \,\,}\|\zeta\|\leq p^{k},\\ &\\ 0&\text{if \,\,}\|\zeta\|>p^{k}.\end{array}\right.$$

$${\int }_{{\mathbb{Z}}_p}{\mathcal{X}}_{D(a,1/p^k)}(x){\zeta }^{-x}d\mu (x)$$

$$f(x,y)=\frac{\left|\{{\alpha }_1,{\alpha }_2,...,{\alpha }_N\}\cap D(x,|y{|}_p)\right|}{N}-|y{|}_p,$$

$$D_N={\sup }_{x,y\in {\mathbb{Z}}_p}\left|f(x,y)\right|.$$

$$f(x,y)$$

$$D_N^4\le C_1(p)\Vert f{\Vert }_2^2,$$

$$\Vert f{\Vert }_2^2\le C_2(p)\sum _{\zeta \in \mathbb{Z}(p^{\infty })\backslash \{1\}}\frac{1}{\Vert \zeta {\Vert }^3}{\left|\frac{1}{N}\sum _{n=1}^N{\zeta }^{x_n}\right|}^2.$$

$$\lfloor x\rfloor =\max \{p^k|k\in \mathbb{Z},p^k\le x\}$$

$$\lceil x\rceil =\min \{p^k|k\in \mathbb{Z},x\le p^k\}.$$

$$f(x,y)$$

$$\Vert f{\Vert }_2^2$$

$$f(x,y)$$

$$\Vert f{\Vert }_2^2$$

$$\Vert f{\Vert }^2\ge \frac{(p-1{)}^3}{p^9}{\Delta }^4$$

Full text

1. Introduction

The theory of equidistribution of sequences modulo one was initiated by Hermann Weyl in $1916$. Since then, it has spurred a lot of interest in many areas of mathematics, including number theory, harmonic analysis, and ergodic theory. The standard reference in this subject is Kuipers and Niederreiter [7].

Equidistribution of sequences on the ring of $p$-adic integers was previously studied in [1, 2, 3]. In particular, Cugiani in [3] defines equidistribution and shows that the sequence $na+b$ is equidistributed if $a$ is a unit. Beer does a quantitative analysis in [1] and [2]. Our aim is to derive a LeVeque-type inequality on the discrepancy of a finite sequence using Fourier analysis.

Let $|\cdot|_{p}$ denote the $p$-adic absolute value on ${\mathbb{Q}}_{p}$, and let

[TABLE]

be the ring of $p$-adic integers. Any element of ${\mathbb{Z}}_{p}$ can be given a unique canonical expansion of the form $x=a_{0}+a_{1}p+a_{2}p^{2}+....$, where the $a_{i}$ are elements of $\{0,1,2,...,p-1\}$ (see for example [5, 6]).

For $k\geq 0$, and $a\in{\mathbb{Z}}_{p}$, we denote by

[TABLE]

a disc of radius $1/p^{k}$ centered at $a$. Note that ${\mathbb{Z}}_{p}$ can be written as the union of $p^{k}$ disjoint discs of the form

[TABLE]

Hence, it is natural to define a notion of equidistribution using such sets.

Definition 1**.**

A sequence $\{\alpha_{n}\}$ is said to be equidistributed in ${\mathbb{Z}}_{p}$ if for every $a$ in ${\mathbb{Z}}_{p}$ and every $k\in{\mathbb{N}}$, we have

[TABLE]

That is, the proportion of the first $N$ elements of $\{\alpha_{n}\}$ lying in a disc $D(a,1/p^{k})$ is equal to its measure in the limit of large $N$, and this holds true for all such discs.

This definition of equidistribution in ${\mathbb{Z}}_{p}$ was first given by Cugiani in [3], where propositions 4 and 10 were also proved. The details are also given in Kuipers and Niederreiter [7]. One also wants to measure how well a sequence distributes itself. To this end, we define the notion of discrepancy to quantify the idea that some sequences are better equidistributed than others.

Definition 2**.**

The discrepancy of a finite sequence $\{\alpha_{1},\alpha_{2},...,\alpha_{N}\}$ in ${\mathbb{Z}}_{p}$ is

[TABLE]

Some elementary arguments show that

[TABLE]

The main aim of this paper is to prove a Fourier analytic upper bound on the discrepancy of a set of $N$ elements $\{\alpha_{1},\alpha_{2},...,\alpha_{N}\}$ in ${\mathbb{Z}}_{p}$.

Let ${\mathbb{Z}}(p^{\infty})$ denote the Prüfer $p$-group, the group of all $p$-th power roots of unity in ${\mathbb{C}}$. Suppose that $\zeta\in{\mathbb{Z}}(p^{\infty})$ has order $p^{n}$, and let $x\in{\mathbb{Z}}_{p}$ have the canonical expansion $x=a_{0}+a_{1}p+a_{2}p^{2}+....+a_{n}p^{n}+.....$. Then we interpret the notation $\zeta^{x}$ as

[TABLE]

Every element of ${\mathbb{Z}}(p^{\infty})$ has finite order, and we denote the order of $\zeta\in{\mathbb{Z}}(p^{\infty})$ by $\|\zeta\|$.

Theorem 1** (Main Theorem).**

The discrepancy of a finite sequence $\{\alpha_{1},...,\alpha_{N}\}$ in ${\mathbb{Z}}_{p}$ is bounded by

[TABLE]

where $C(p)$ is a constant dependent on $p$.

As an example application of Theorem 1, we have the following corollary

Corollary 2**.**

The sequence $na+b$ where $a$ is a unit in ${\mathbb{Z}}_{p}$ has discrepancy

[TABLE]

Some quantitative results on the discrepancy of $p$-adic sequences were done by Beer in [1] and [2]. In particular, the author proves in [1] that the discrepancy of the sequence $na+b$ with $a$ a unit is exactly equal to $D_{N}=N^{-1}$, the best possible.

It is not surprising that the LeVeque type inequality gives us a weaker bound, as this is the case in the classical setting on ${\mathbb{R}}/{\mathbb{Z}}$. Montogomery in [8] provides a detailed discussion and considers some examples. In particular, the sequence $n\theta$ where $\theta=\frac{1+\sqrt{5}}{2}$ has discrepancy $D_{N}\ll\log(N)/N$, where as the use of the LeVeque inequality gives only $D_{N}\ll N^{-2/3}$.

Our paper is structured as follows. In section 2 we set up the relevant Fourier analysis that is required for our calculations. We prove the main theorem in section 3. We analyze the quantitative behavior of the linear sequence $\alpha_{n}=na+b$ in section 4, and prove Corollary 2.

2. Fourier analysis on ${\mathbb{Z}}_{p}$

If $G$ is a compact abelian group, then the set of all continuous group homomorphisms (or characters) from $G$ to the multiplicative unit circle ${\mathbb{T}}=\{z\in{\mathbb{C}}\,|\,|z|=1\}$ forms a discrete group under multiplication, the Pontyagrin dual group $\widehat{G}$ (see for example [9]). Note that ${\mathbb{Z}}_{p}$ is a compact abelian group. The next lemma states that the dual group $\widehat{{\mathbb{Z}}}_{p}$ of ${\mathbb{Z}}_{p}$ is isomorphic to the Prüfer $p$-group ${\mathbb{Z}}(p^{\infty})$. The result is known, but we include a proof due to the lack of a suitable reference.

Lemma 3**.**

For each $\zeta\in{\mathbb{Z}}(p^{\infty})$, the map $x\mapsto\zeta^{x}$ is a character of ${\mathbb{Z}}_{p}$. Moreover, the map

[TABLE]

is an isomorphism from the Prüfer $p$-group ${\mathbb{Z}}(p^{\infty})$ to the Pontyagrin dual group of ${\mathbb{Z}}_{p}$.

Proof.

It is easily shown that the map $x\mapsto\zeta^{x}$ is a character of ${\mathbb{Z}}_{p}$. To show the injectivity of $\Psi$, suppose that $\zeta_{1}^{x}=\zeta_{2}^{x}$ for all $x$ in ${\mathbb{Z}}_{p}$. Then picking $x=1$, we get $\zeta_{1}=\zeta_{2}$.

We need argue that $\Psi$ is surjective. Let $\gamma$ be in the dual group of ${\mathbb{Z}}_{p}$. Since $\gamma(0)=1$ and $\gamma$ is continuous, there exists a disc of radius $1/p^{n}$ centered at zero $D=p^{n}{\mathbb{Z}}_{p}$, such that $|\gamma(x)-\gamma(0)|<1$ for all $x$ in $D$, and we can pick a smallest $n$ such that this is true. Moreover, since $D$ is a subgroup of ${\mathbb{Z}}_{p}$ we must have that the image $\gamma(D)$ is a subgroup of ${\mathbb{T}}$.

Note that there does not exist any non-trivial subgroup of ${\mathbb{T}}$ satisfying the condition $|x-y|<1$ for all elements $x$ and $y$ in the subgroup. Hence, we conclude that $\gamma(D)=\{1\}$.

Now suppose that $\gamma(1)=\zeta=e^{2\pi i\theta}$ for some $\theta$ in $[0,1)$. Then, $\gamma(p^{n})=\gamma(1)^{p^{n}}=e^{2\pi ip^{n}\theta}=1$ or $p^{n}\theta\in{\mathbb{Z}}$. We conclude that $\theta=\frac{m}{p^{n}}$, where $p\not|\hskip 2.0ptm$ by the minimality of $n$.

For any integer value $k$, we have $\gamma(k)=\gamma(1)^{k}=\zeta^{k}$. This completely determines $\gamma$, since we can write ${\mathbb{Z}}_{p}$ as the union of $p^{n}$ disjoint balls ${\mathbb{Z}}_{p}=\cup_{k=0}^{p^{n}-1}\left(k+p^{n}{\mathbb{Z}}_{p}\right)$ and for any $x\in{\mathbb{Z}}_{p}$ we have $x=k+p^{n}y$, and $\gamma(x)=\gamma(k)$. We conclude that

[TABLE]

for all $x$ in ${\mathbb{Z}}_{p}$ where $\zeta=e^{\frac{2\pi im}{p^{n}}}$. ∎

Using Lemma  3, we shall express the Fourier series of any $f\in L^{1}({\mathbb{Z}}_{p})$ in terms of the elements of ${\mathbb{Z}}(p^{\infty})$. As a compact group there exists a normalized Haar measure $\mu$ on ${\mathbb{Z}}_{p}$ (see for example [4]). Let $f\in L^{1}({\mathbb{Z}}_{p})$. The Fourier coefficients of $f$ are given by

[TABLE]

and the Fourier inversion formula gives

[TABLE]

whenever $\sum_{\zeta\in{\mathbb{Z}}(p^{\infty})}|\hat{f}(\zeta)|<\infty$.

A Weyl type criterion holds for equidistribution in ${\mathbb{Z}}_{p}$. We state it here in terms of the elements of ${\mathbb{Z}}(p^{\infty})$, although it holds for a more general class of Riemann integrable functions on ${\mathbb{Z}}_{p}$ (see [7]).

Proposition 4** (Weyl’s Criterion).**

A sequence $\{\alpha_{n}\}$ is equidistributed in ${\mathbb{Z}}_{p}$ if and only if for every non-trivial $\zeta$ in ${\mathbb{Z}}(p^{\infty})$ we have

[TABLE]

We denote by ${\mathcal{X}}_{D(a,R)}(x)$ the characteristic function of the disc $D(a,R)$ centered at $a$ of radius $R$. We have the following change of variables formula, the proof of which is elementary and we omit.

Proposition 5**.**

Let $f:{\mathbb{Z}}_{p}\to{\mathbb{C}}$ be an integrable function. Then

[TABLE]

We use Proposition 5 to calculate the Fourier coefficients of the characteristic function of a disc.

Lemma 6**.**

The Fourier coefficients of the characteristic function ${\mathcal{X}}_{D(a,1/p^{k})}(x)$ are

[TABLE]

Proof of Lemma  6.

Suppose that $\|\zeta\|\leq p^{k}$, then $\zeta^{p^{k}x}=1$ for all $x$ in ${\mathbb{Z}}_{p}$. Therefore, we have

[TABLE]

On the other hand suppose $\|\zeta\|>p^{k}$, and let $\omega=\zeta^{p^{k}}$. Then $\|\omega\|=\|\zeta\|/p^{k}>1$ and hence

[TABLE]

∎

3. Proof of the main theorem

Let $\{\alpha_{1},\alpha_{2},...,\alpha_{N}\}$ be a finite sequence in ${\mathbb{Z}}_{p}$. Define the function $f:{\mathbb{Z}}_{p}\times{\mathbb{Z}}_{p}\longrightarrow{\mathbb{R}}$

[TABLE]

where $D(x,|y|_{p})$ is a disc of radius $|y|_{p}$ centered at $x$. The discrepancy of the points $\{\alpha_{1},...,\alpha_{N}\}$ is then

[TABLE]

We suppress the $p$ in $|\cdot|_{p}$ as it would be clear from the context. We can also write

[TABLE]

Our proof of Theorem 1 proceeds as follows. We shall bound the $L^{2}$ norm $\displaystyle\|f\|_{2}^{2}=\iint_{{\mathbb{Z}}_{p}^{2}}|f(x,y)|^{2}\,d\mu(x)d\mu(y)$ from below by $D_{N}^{4}$ using geometrical arguments, and from above by using Parseval’s theorem. The two steps are given below as lemmas

Lemma 7**.**

The discrepancy $D_{N}$ is bounded by

[TABLE]

where $C_{1}(p)$ is a constant dependent on $p$.

Lemma 8**.**

The $L^{2}$ norm of the function $f$ is bounded by

[TABLE]

where $C_{2}(p)$ is a constant dependent on $p$.

The proof of Theorem 1 then follows by combining Lemmas 7 and 8.

Remark 1**.**

For $x>0$, we use the notation $\lfloor x\rfloor$ and $\lceil x\rceil$ to denote

[TABLE]

[TABLE]

Note that $\lfloor x\rfloor\leq x<p\lfloor x\rfloor$ and $\frac{1}{p}\lceil x\rceil<x\leq\lceil x\rceil$.

Proof of Lemma  7.

Pick a point $(x_{0},y_{0})$ for which $f(x_{0},y_{0})$ is not zero. We consider each of the two possibilities $f(x_{0},y_{0})>0$ and $f(x_{0},y_{0})<0$ separately. Our strategy in each case is to find a small neighborhood around the point $(x_{0},y_{0})$ where $|f(x,y)|$ is bounded away from zero. Using this fact and integrating over this neighborhood, we produce a bound of the form $\|f\|_{2}^{2}\geq C(p)|f(x_{0},y_{0})|^{4}$, where $C(p)$ is a constant depending only on $p$.

Case 1

Suppose that $\Delta=f(x_{0},y_{0})>0.$ Let $R=\lfloor\Delta+|y_{0}|\rfloor$. Since, $|y_{0}|<|y_{0}|+\Delta$ and $|y_{0}|$ is in the value group of ${\mathbb{Q}}_{p}$, we have $|y_{0}|\leq R$. We consider the two cases $|y_{0}|<R$ and $|y_{0}|=R$.

Case 1.1:

Suppose that $|y_{0}|<R$. We must then have $|y_{0}|\leq\frac{1}{p}R$. If we fix $|y|=\frac{1}{p}R$ and $|x-x_{0}|\leq\frac{1}{p}R$, then $D(x_{0},|y_{0}|)\subseteq D(x,|y|)$. We get a nonnegative lower bound on $f(x,y)$ as follows

[TABLE]

We can bound the $L^{2}$ norm of $f$ from below by evaluating the required integral only on the set $|y|=\frac{1}{p}R,|x-x_{0}|\leq\frac{1}{p}R$

[TABLE]

using $R=\lfloor\Delta+|y_{0}|\rfloor\geq\frac{1}{p}(|y_{0}|+\Delta)\geq\frac{1}{p}\Delta$.

Case 1.2:

Suppose that $|y_{0}|=R$. If we let $|y|=R$ and $|x-x_{0}|\leq R$, then $D(x_{0},|y_{0}|)=D(x,|y|)$. From this, we get

[TABLE]

Therefore,

[TABLE]

using $R+\Delta=|y_{0}|+\Delta<p\lfloor|y_{0}|+\Delta\rfloor=pR$ and therefore $\Delta<(p-1)R$.

Finally, since $\frac{(p-1)^{3}}{p^{9}}<\frac{1}{p(p-1)}$ we conclude

[TABLE]

holds in both cases 1.1 and 1.2, so it holds in general for case 1.

Case 2:

Suppose that $f(x_{0},y_{0})<0$ and $\displaystyle\Delta=|f(x_{0},y_{0})|=-f(x_{0},y_{0}).$ In other words, the disc $D(x_{0},|y_{0}|)$ contains fewer than the expected number of points $\alpha_{n}$.

Now let $R=|y_{0}|$. Then if $|y|=R$ and $|x-x_{0}|\leq R$, by the strong triangle inequality $D(x,|y|)=D(x_{0},|y_{0}|)$ and we have

[TABLE]

Therefore,

[TABLE]

where the last line follows because $\Delta\leq R$. To see this, note that

[TABLE]

∎ Next, we need to prove Lemma 8. Our goal is to find an upper bound on the $L^{2}$-norm of $f(x,y)$ using Parseval’s theorem. Suppose $f(x,y)$ has a Fourier series

[TABLE]

Then by Parseval’s theorem we would get

[TABLE]

Therefore, we need to bound the Fourier coefficients of $f(x,y)$. The Fourier coefficients are

[TABLE]

Note that if $\zeta=1$ we get

[TABLE]

When $\zeta\neq 1$, the second integral in line 2 of Equation (1) is zero

[TABLE]

Therefore,

[TABLE]

Using Lemma 6, we have

[TABLE]

Hence, for $\zeta\neq 1$,

[TABLE]

The following lemma makes some estimates that are useful in our succeeding calculations

Lemma 9**.**

Let $R=p^{k},k\in{\mathbb{Z}}$ satisfy $0<R<1$, and let $\omega\in{\mathbb{Z}}(p^{\infty}).$ Then,

[TABLE]

Moreover,

[TABLE]

Proof of Lemma 9.

Let $R=1/p^{k}$ and let $\|\omega\|=p^{l}$. We have

[TABLE]

When $\|\omega\|\leq 1/R$, that is when $l\leq k$, using Lemma 6 and (9) we have

[TABLE]

Thus (7) holds in this case. If $\|\omega\|>1/R$, that is when $l\geq k+1$, again using Lemma 6 and (9) we have

[TABLE]

and thus (7) holds in this case as well.

To check (8) , we use the fact that for each $j\geq 1$, the group ${\mathbb{Z}}(p^{\infty})$ contains $p^{j}$ elements of order at most $p^{j}$, and $p^{j}-p^{j-1}$ elements of order exactly $p^{j}$. We then have

[TABLE]

∎

Finally, we prove Lemma 8.

Proof of Lemma 8.

Applying Parseval’s theorem to $f(x,y)$ and using Equations (4) and (8), we conclude

[TABLE]

∎

4. The linear sequence $na+b$ in ${\mathbb{Z}}_{p}$

Consider the sequence $\alpha_{n}=na+b$. We have the following proposition, a proof of which is given in [7] using elementary number theory. We present an alternate proof using Fourier analysis.

Proposition 10**.**

The sequence $\alpha_{n}=na+b$ is equidistributed in ${\mathbb{Z}}_{p}$ if and only if $a$ is a unit in ${\mathbb{Z}}_{p}$.

Proof.

The forward implication follows from Weyls criterion (Proposition 4). For suppose, $a$ was not a unit. Then $a=p^{k}c$, where $k>0$ and $c$ is a unit. Now let $\zeta=e^{2\pi i/p^{k}}$. Then $\zeta^{a}=1$, and Weyl’s criterion will not hold.

For the reverse implication, let $\zeta\in{\mathbb{Z}}(p^{\infty})$ with $\|\zeta\|=p^{k}$ for $k\geq 1$. There exists an $m$ such that $1\leq m<p^{k}$, with $p\not|\hskip 2.0ptm$ and $\displaystyle\zeta=e^{2\pi im/p^{k}}$. Suppose that $a$ is a unit in ${\mathbb{Z}}_{p}$. Let $a=t_{0}+t_{1}p+t_{2}p^{2}+...$ be the canonical expansion of $a$, with $t_{0}\neq 0$. Then we let $a_{k}=t_{0}+t_{1}p+...+t_{k-1}p^{k-1}$ be the truncation of this expansion to the first $k$ terms. We have

[TABLE]

Since $p\not|\hskip 2.0ptm$ and $p\not|\hskip 2.0pta_{k}$, $\displaystyle\sin(\pi ma_{k}/p^{k})\neq 0$ and hence $\frac{1}{N}\sum_{n=1}^{N}\zeta^{na+b}\to 0$ as $N\to\infty$; the proof of equidistribution now follows from Weyl’s criterion. ∎

Proof of Corollary 2.

Applying the bound given by Theorem 1 we get

[TABLE]

Note that the second inequality in Equation (17) comes from the last inequality in (15). For the fourth inequality, note that since $a$ is a unit we have $p\not|\hskip 2.0pta_{k}$. Hence, $\gcd(a_{k},p^{k})=1$ and so $a_{k}$ generates ${\mathbb{Z}}/p^{k}{\mathbb{Z}}$. That is, ${\mathbb{Z}}/p^{k}{\mathbb{Z}}=\left\{ma_{k}\,|\,m=0,..,p^{k}-1\right\}$. The final inequality follows from the identities $|\sin(\theta)|=|\sin(-\theta)|=|\sin(\pi-\theta)|$, so that for $p^{k}/2\leq l<p^{k}$ we have $|\sin(\pi l/p^{k})|=|\sin(\pi(p^{k}-l)/p^{k})|$. This allows us to double the sum over the first half of the interval.

Note that in the interval $[0,\,\pi/2]$, $\sin(\theta)$ is bounded from below by $2\theta/\pi$, so that

[TABLE]

This gives us

[TABLE]

Finally, applying the bound from (26) to (17) we get

[TABLE]

We conclude that $D_{N}=O\left(\frac{1}{\sqrt{N}}\right)$. ∎