Discrepancy estimates for generalized polynomials

Abstract

We obtain an upper bound for the discrepancy of the sequence $([p(n)\alpha ]\beta )_{n\ge 0}$ generated by the generalized polynomial $[p(x)\alpha ]\beta$, where p(x) is a monic polynomial with real coefficients, $\alpha$ and $\beta$ are irrational numbers satisfying certain conditions.

1 Introduction

A sequence $(x_n)_{n\ge 0}$ of real numbers is said to be uniformly distributed modulo 1 if

$$\begin{aligned} \lim _{N\rightarrow \infty }\frac{\#\{n\le N: \{x_n\}\in [a,b)\}}{N}\; =\; b-a \end{aligned}$$

(1)

holds for all real numbers a, b satisfying $0\le a<b\le 1$. Here and in what follows, $\{x\}$ denotes the fractional part of x. Weyl [10] proved that if $P(x)\in \mathbb {R}[x]$ is any polynomial in which at least one of the coefficients other than the constant term is irrational, then the sequence $(P(n))_{n\ge 0}$ is uniformly distributed modulo 1.

A natural extension of the family of real valued polynomials arises by adding the operation integral part, denoted by $[\cdot ]$, to the arithmetic operations addition and multiplication. Polynomials which can be obtained in this way are called generalized polynomials. For example $[a_0+a_1x]$, $a_0+[a_1x+[a_2x^2]]$ are generalized polynomials.

In the spirit of Weyl’s result it is natural to consider the uniform distribution of generalized polynomials. The case $([n\alpha ]\beta )_{n\ge 0}$ is treated in [8] (see Theorem 1.8, p. 310) and it follows from a result of Veech (see Theorem 1, [9]) that the sequence $([p(n)]\beta )_{n\ge 0}$, p(x) is a polynomial with real coefficients, is uniformly distributed under certain conditions on the coefficients of p(x) and $\beta$. Håland [4, 5] showed that if the coefficients of a generalized polynomial q(x) are sufficiently independent then the sequence $(q(n))_{n\ge 0}$ is uniformly distributed.

In order to quantify the convergence in (1) the notion of discrepancy has been introduced. Let $(x_n)_{n\ge 0}$ be a sequence of real numbers and N be any positive integer. The discrepancy of this sequence, denoted by $D_N(x_n)$, is defined by

$$\begin{aligned} D_N(x_n)\;=\; \sup _{0\le a<b\le 1}\left| \frac{\#\{n\le N:\{x_n\}\in [a, b)\}}{N}-(b-a)\right| . \end{aligned}$$

Now we have the following definition.

Definition 1

Let $t\ge 1$ be a real number. We say that a pair $(\alpha , \beta )$ of real numbers is of finite type t if for each $\epsilon >0$ there is a positive constant $c=c(\epsilon ,\alpha ,\beta )$ such that for any pair of rational integers $(m,n)\not =(0,0)$, we have

$$\begin{aligned} (\max (1,|m|))^{t+\epsilon }(\max (1,|n|))^{t+\epsilon }\Vert m\alpha +n\beta \Vert \ge c \end{aligned}$$

where $\Vert x\Vert$ denotes the distance of x from the nearest integer.

The corresponding definition for a single real number $\alpha$ is the one of irrationality measure. The precise definition is the following.

Definition 2

Let $t\ge 1$ be a real number. We say that an irrational number $\gamma$ has irrationality measure $t+1$ if for any integer n and $\epsilon >0$, we have

$$\begin{aligned} \max (1,|n|)^{t+\epsilon }\Vert n\gamma \Vert \gg _{\epsilon ,\gamma } 1. \end{aligned}$$

It is well known that when $\gamma$ has irrationality measure $t+1$, the discrepancy $D_N(n\gamma )$ of the sequence $(n\gamma )_{n\ge 0}$ satisfies

$$\begin{aligned} D_N(n\gamma )\ll _{\gamma ,\epsilon }N^{\frac{-1}{t}+\epsilon } \end{aligned}$$

for each $\epsilon >0$.

The discrepancy of non-trivial generalized polynomials was first considered by Hofer and Ramaré [6]. More precisely, they considered the discrepancy of the sequence $([n\alpha ]\beta )_{n\ge 0}$ and proved that for each $\epsilon >0$

$$\begin{aligned} D_N([n\alpha ]\beta )\ll _{\epsilon ,\alpha ,\beta } N^{\frac{-1}{3t-2}+\epsilon } \end{aligned}$$

when $(\alpha , \alpha \beta )$ and $(\beta , \frac{1}{\alpha })$ are of finite type t.

Let $p(x)=x^d+a_{d-1}x^{d-1}+\cdots +a_1x+a_0\in \mathbb {R}[x]$ be a monic polynomial of degree $d\ge 2$. In this paper we consider the discrepancy of the sequence $([p(n)\alpha ]\beta )_{n\ge 0}$. We prove the following theorem.

Theorem 1

Let $\alpha$, $\beta$ and $N>1$ be non-zero real numbers. Suppose that the pair $(\alpha ,\alpha \beta )$ is of finite type t for a real number $t \ge 1$. Then for any $\epsilon >0$,

$$\begin{aligned} D_N([p(n)\alpha ]\beta ) \ll _{\epsilon ,\alpha ,\beta ,d} N^{-\frac{2-2^{-d+2}}{2^{d-1}(2t+1)+7t+2}+\epsilon }. \end{aligned}$$

We use a modified version of the method of Hofer and Ramaré [6] for the proof of the above theorem.

Remark 1

The above theorem, in particular, shows that the sequence $([p(n)\alpha ]\beta )$ is uniformly distributed if $(\alpha ,\alpha \beta )$ is of finite type t for $t\ge 1$. This fact also follows from a theorem of Carlson (see Theorem 2, [2]). Theorem 1 of [9] also implies uniform distribution of this sequence under certain conditions on the coefficients of the polynomial p(x).

2 Preliminaries

For any real number $\tau$, let $f_{\tau }(x)=e(\tau \{ x \} )$ where e(x) denotes $e^{2\pi ix}$. Let $\delta >0$ be a real number. We are going to approximate $f_\tau$ by a function $g_{\tau ,\delta }$. Here $g_{\tau ,\delta }$ is defined by

$$\begin{aligned} g_{\tau , \delta }(x)= \frac{1}{ (2\delta )^r} 1_{[-\delta , \delta ]}*\cdots *1_{[-\delta , \delta ]}*f_\tau (x), \end{aligned}$$

(2)

where we have r copies of $1_{[-\,\delta , \delta ]}$ each denoting the indicator function of the interval $[-\,\delta , \delta ]$.

We have the following analog of Lemma 3.1 in [6].

Lemma 1

For any sequence $\{u_n\}_{n\ge 0}$ of real numbers, and any positive integer N we have

$$\begin{aligned} \sum _{n=0}^{N-1} | f_{\tau }(u_n) - g_{\tau ,\delta }(u_n)| \ll Nr\delta + Nr^2 \delta |\tau | + ND_N(u_n). \end{aligned}$$

(3)

Using Fourier inversion formula, we have

$$\begin{aligned} g_{\tau , \delta }(x)\;=\; \sum _{k\in \mathbb {Z}}\hat{g}_{\tau , \delta }(k)e(-kx)\; , \end{aligned}$$

with

$$\begin{aligned} \hat{g}_{\tau , \delta }(k)\;=\; \left( \frac{\sin 2\pi k\delta }{2\pi k\delta }\right) ^r\frac{e(\tau +k)-1}{2\pi i(k+\tau )}. \end{aligned}$$

Since $\left| \frac{\sin 2\pi x}{x}\right| ^r\ll _r \min \left( 1,\frac{1}{|x|^r}\right)$, and for any irrational $\tau$, $|e(\tau )-1|\ll \Vert \tau \Vert$, we have the following lemma which holds trivially.

Lemma 2

For any irrational number $\tau$, we have

$$\begin{aligned} |\hat{g}_{\tau , \delta }(k)| \ll _r \frac{\Vert \tau +k\Vert }{|\tau +k|}\min \left( 1, \frac{1}{(|k|\delta )^r} \right) . \end{aligned}$$

We will state Lemmas 3 and 4 for arbitrary real number $\tau$ but we keep in mind that we will use these lemmas with $\tau =-\,h\beta$, for some positive integer h. The next lemma gives an upper bound for the tail of the Fourier series of $g_{\tau ,\delta }$.

Lemma 3

Let K be sufficiently large real number such that $|\tau +k|\ge \frac{k}{2}$ for all $k\in \mathbb {Z}$ with $|k|>K$. Then we have

$$\begin{aligned} \sum _{ |k| > K } \hat{g}_{\tau , \delta }(k)\ll _r (\delta K)^{-r}. \end{aligned}$$

The following lemma shows that for any $p>1$ the $L^p$-norm of $\hat{g}_{\tau , \delta }$ is bounded.

Lemma 4

Let $\tau$ be a real number and $0<\delta < \min \left( \frac{1}{2|\tau |},1\right)$. Then for any real number $p>1$, we have

$$\begin{aligned} \sum _{k\in \mathbb {Z}} |\hat{g}_{\tau , \delta }(k)|^p \ll 1\; , \end{aligned}$$

where the implied constant depends only on p.

Proof

We can assume the sum is running over $k\ge 1$. Using Lemma 2, we get

$$\begin{aligned} \sum _{k\ge 1} |\hat{g}_{\tau , \delta }(k)|^p\le & {} \sum _{k\ge 1} \frac{||\tau +k||^p}{|\tau +k|^p} \min \left( 1, \frac{1}{(k\delta )^{pr}}\right) \\= & {} \sum _{k\le \delta ^{-1}} \frac{||\tau +k||^p}{|\tau +k|^p} +\delta ^{-pr}\sum _{k>\delta ^{-1}} \frac{1}{|\tau +k|^p k^{pr}}\;. \end{aligned}$$

Note that $k> \delta ^{-1} >2|\tau |$ implies $|\tau +k| \ge k/2$, hence

$$\begin{aligned} \sum _{k>\delta ^{-1}} \frac{1}{|\tau +k|^p k^{pr}} \ll \delta ^{p(r+1)-1} \;. \end{aligned}$$

Hence we have

$$\begin{aligned} \sum _{k\ge 1} |\hat{g}_{\tau , \delta }(k)|^p \ll \sum _{k\le \delta ^{-1}} \frac{||\tau +k||^p}{|\tau +k|^p}+ 1. \end{aligned}$$

When $\tau$ is a non-negative real number, sum on the right hand side is clearly $\ll 1$. Hence we can assume that $\tau$ is a negative real number. The contributions for the sum above from the terms with $k=[-\,\tau ]$ and $k=[-\,\tau ]+1$ are $\le 1$. Hence we have

$$\begin{aligned} \sum _{k\ge 1} |\hat{g}_{\tau , \delta }(k)|^p \ll S_1+S_2+1 \; , \end{aligned}$$

where

$$\begin{aligned} S_1 = \sum _{k=1}^{[-\,\tau ]-1} \frac{1}{|\tau +k|^p} \ \text { and } \ S_2=\sum _{k=[-\,\tau ]+2}^{\delta ^{-1}}\frac{1}{|\tau +k|^p}. \end{aligned}$$

Now the summand in $S_1$ is monotonically increasing, hence

$$\begin{aligned} S_1= \int _1^{[-\,\tau ]-1}\frac{dx}{(\tau +x)^p} +O\left( \frac{1}{(\tau +[-\,\tau ]-1)^p}\right) +O\left( \frac{1}{(\tau +1)^p}\right) . \end{aligned}$$

It is easy to see that

$$\begin{aligned} \int _1^{[-\,\tau ]-1}\frac{dx}{(\tau +x)^p} \ll 1, \end{aligned}$$

as $p>1.$ Thus we conclude

$$\begin{aligned} S_1\ll 1 . \end{aligned}$$

In a similar way, with only difference being the summand is monotonically decreasing, one can show that

$$\begin{aligned} S_2\ll 1 \end{aligned}$$

which finishes the proof. $\square$

Now we need a variant of a lemma of Weyl–van der Corput (see Lemma 2.7, [1]) as given by Granville and Ramaré ( see Lemma 8.3 of [3]).

Lemma 5

Suppose that $\lambda _1,\lambda _2,\ldots ,\lambda _N$ is a sequence of complex numbers, each with $|\lambda _i|\le 1$, and define $\Delta \lambda _m=\lambda _m$, $\Delta _r\lambda _m=\lambda _{m+r}\overline{\lambda }_m$ and

$$\begin{aligned} \Delta _{r_1,\ldots ,r_k,s}\lambda _m\;=\; (\Delta _{r_1,\ldots ,r_k}\lambda _{m+s})\overline{(\Delta _{r_1,\ldots ,r_k} \lambda _m)}. \end{aligned}$$

Then for any given $k\ge 1$, and real number $Q\in [1,N]$,

$$\begin{aligned} \left| \frac{1}{8N}\sum _{m=1}^N\lambda _m\right| ^{2^k}\le \frac{1}{8Q}+\frac{1}{8Q^{2-2^{-k+1}}}\sum _{r_1=1}^{Q}\sum _{r_2=1}^{Q^{\frac{1}{ 2}}}\cdots \sum _{r_k=1}^{Q^{2^{-k+1}}}\left| \frac{1}{N}\sum _{m=1}^{ N-r_1-\cdots -r_k}\Delta _{r_1,\ldots ,r_k}\lambda _m\right| . \end{aligned}$$

The following lemma, often called as Erdős–Turán inequality, is very useful to estimate the discrepancy of a given sequence (see Theorem 2.5, p. 112 of [8]).

Lemma 6

(Erdős–Turán) Let $(x_n)_{n\ge 0}$ be any sequence of real numbers and $N\ge 1$. The discrepancy $D_N(x_n)$ of the sequence $(x_n)_{n\ge 0}$ satisfies the following:

$$\begin{aligned} D_N(x_n)\le \frac{6}{H+1}+\frac{4}{\pi }\sum _{h=1}^H \frac{1}{h}\left| \frac{1}{N}\sum _{n=0}^{N-1}e(hx_n)\right| , \end{aligned}$$

(4)

where H is any arbitrary positive integer.

The above lemma shows that the exponential sums play an important role not only in showing the uniform distribution of a sequence, but also in estimating the discrepancy of a given sequence.

The following lemma is an easy consequence of Lemma 6.

Lemma 7

Let $\theta$ be an irrational number. Then the discrepancy $D_L(\ell \theta )$ of the sequence $\{\ell \theta : 1\le \ell \le L\}$ satisfies the following upper bound.

$$\begin{aligned} D_L(\ell \theta ) \le C\left( \frac{1}{H} + \frac{1}{L} \sum _{j=1}^H \frac{1}{j\Vert j\theta \Vert } \right) \end{aligned}$$

for any $H>1$ and for some absolute constant $C>0$.

If $\alpha$ is of irrationality measure $t+1$ for $t\ge 1$, it is known that the discrepancy of $(n^2\alpha )$ satisfies the following upper bound.

$$\begin{aligned} D_N(n^2\alpha )\ll _{\epsilon ,t} N^{-\frac{1}{t+1} +\epsilon }+N^{-\frac{2}{5}}\sqrt{\log N} \end{aligned}$$

for any $\epsilon >0$ (see equation (50) p. 113 in [7]). To estimate the discrepancy of $([p(n)\alpha ]\beta )_{n\ge 0}$, we need the following general version.

Proposition 1

Let $\alpha$ be a non-zero real number of irrationality measure $t+1$ for a real $t\ge 1$. Then the discrepancy $D_N(p(n)\alpha )$ of the sequence $(p(n)\alpha )_{n\ge 0}$ satisfies

$$\begin{aligned} D_N(p(n)\alpha )\ll _{\epsilon ,d,t} N^{-\frac{2-2^{-d+2}}{2^{d-1}(2t+1)}+\epsilon } \end{aligned}$$

for any $\epsilon >0$.

Proof

Let $x_n=p(n)\alpha$ in Lemma 6. Then

$$\begin{aligned} D_N(p(n)\alpha ) \ll \frac{1}{H}+ \frac{1}{N} \sum _{h=1}^H\frac{1}{h}\left| \sum _{n=0}^{N-1} e(p(n)h\alpha )\right| . \end{aligned}$$

(5)

To estimate the exponential sum on the right hand side we use Lemma 5 with $Q=N$ and $k=d-1$. Hence we get that

$$\begin{aligned} \begin{aligned}&\left| \sum _{n=0}^{N-1} e(p(n)h\alpha )\right| ^{2^{d-1}} \ll N^{2^{d-1}-1} \\&\quad +N^{2^{d-1}+2^{-d+2}-3}\sum _{r_1=1}^N\cdots \sum _{r_{d-1}=1}^{N^{2^{ -d+2}}}\left| \sum _{n=0}^{N-r_1-\cdots -r_{d-1}}e(d!hr_1\cdots r_{d-1}n\alpha )\right| . \end{aligned} \end{aligned}$$

Using the bound $|\sum _{n=0}^{N-1}e(n\lambda )|\ll \min (N,\frac{1}{\Vert \lambda \Vert })$ gives

$$\begin{aligned}&\left| \sum _{n=0}^{N-1} e(p(n)h\alpha )\right| ^{2^{d-1}}\nonumber \\ {}&\quad \ll N^{2^{d-1}-1}+N^{2^{d-1}+2^{-d+2}-3}\sum _{r_1=1}^N\cdots \sum _{r_{d-1}=1}^{N^{2^{-d+2}}}\min \left( N,\frac{1}{\Vert d!hr_1\cdots r_{d-1}\alpha \Vert }\right) \nonumber \\&\quad \ll N^{2^{d-1}-1}+N^{2^{d-1}+2^{-d+2}-3}\sum _{m=1}^{N^{2-2^{-d+2}}} T(m)\min \left( N, \frac{1}{\Vert d!hm\alpha \Vert }\right) , \end{aligned}$$

(6)

where in the second line of the above inequality

$$\begin{aligned} T(m)=\left| \left\{ (r_1,\ldots ,r_{d-1})\in [1,N]\times \cdots \times [1,N^{2^{-d+2}}]: r_1\cdots r_{d-1}=m\right\} \right| . \end{aligned}$$

Hence $T(m)\ll \tau _{d-1}(m)$. Let $\epsilon _1=\frac{\epsilon }{(2-2^{-d+2})}$. Using the fact that $\tau _{d-1}(m)\ll _{\epsilon _1} m^{\epsilon _1}$ we get that

$$\begin{aligned} \begin{aligned}&\left| \sum _{n=0}^{N-1} e(p(n)h\alpha )\right| ^{2^{d-1}} \\&\quad \ll _{\epsilon ,d} N^{2^{d-1}-1}+N^{2^{d-1}+2^{-d+2}-3+\epsilon }\sum _{m=1}^{N^{2-2^{-d+2}}} \min \left( N,\frac{1}{\Vert d!hm\alpha \Vert }\right) . \end{aligned} \end{aligned}$$

(7)

Let $L=N^{2-2^{-d+2}}$. We have

$$\begin{aligned} \sum _{m=1}^L \min \left( N, \frac{1}{\Vert d!mh\alpha \Vert } \right) = N |E_0| + \sum _{m \notin E_0} \frac{1}{\Vert d!mh\alpha \Vert }\; , \end{aligned}$$

where

$$\begin{aligned} E_k=\left\{ m\le L: \frac{k}{N}<\Vert d!mh\alpha \Vert \le \frac{k+1}{N} \right\} . \end{aligned}$$

With this notation we have

$$\begin{aligned} \sum _{m=1}^L \min \left( N, \frac{1}{\Vert d!mh\alpha \Vert } \right) \ll N |E_0| + \sum _{k=1}^{N-1} \frac{N}{k} |E_k|. \end{aligned}$$

Observe that

$$\begin{aligned} |E_k| = \frac{2L}{N} + O( L D_L(d!mh\alpha ) ). \end{aligned}$$

Hence we have

$$\begin{aligned} \sum _{m=1}^L \min \left( N, \frac{1}{\Vert d!mh\alpha \Vert } \right) \ll L \log N + NL D_L(d!mh\alpha )\log N. \end{aligned}$$

(8)

Since $\alpha$ has irrationality measure $t+1$, $\Vert d!mh\alpha \Vert \ge _{\epsilon }(d!mh)^{-(t+\epsilon )}$. Then by Lemma 7

$$\begin{aligned} D_L(d!mh\alpha )&\ll _{\epsilon } \frac{1}{H}+\frac{1}{L}\sum _{j=1}\frac{1}{j\Vert d!hj\alpha \Vert }\\&\ll _{\epsilon ,d,t} \frac{1}{H}+\frac{(d!h)^{t+\epsilon }}{L}\sum _{j=1}^H j^{t-1+\epsilon }\\&\ll _{\epsilon ,d,t} \frac{1}{H}+L^{-1}H^{t+\epsilon }h^{t+\epsilon }. \end{aligned}$$

Choose $H=\left[ L^{\frac{1}{t+1}}h^{-\frac{t}{t+1}}\right]$ to get

$$\begin{aligned} D_L(d!mh\alpha )\ll _{\epsilon ,d,t}L^{-\frac{1}{t+1}+\epsilon }h^{\frac{t}{t+1} +\epsilon }. \end{aligned}$$

(9)

Using this estimate in (8) gives us

$$\begin{aligned} \sum _{m=1}^L \min \left( N, \frac{1}{\Vert d!mh\alpha \Vert } \right) \ll _{\epsilon ,d,t}NL^{1-\frac{1}{t+1}+\epsilon }h^{\frac{t}{t+1}+\epsilon }. \end{aligned}$$

(10)

The above estimate when $L=N^{2-2^{-d+2}}$ together with (7) gives

$$\begin{aligned} \left| \sum _{n=0}^{N-1} e(p(n)h\alpha )\right| ^{2^{d-1}}\ll _{\epsilon ,d,t} N^{2^{d-1}-1}+N^{2^{d-1}-\frac{2-2^{-d+2}}{t+1}+\epsilon }. \end{aligned}$$

(11)

In the above estimate clearly the second term dominates. Hence we get

$$\begin{aligned} \left| \sum _{n=0}^{N-1} e(p(n)h\alpha )\right| \ll _{\epsilon ,d,t} N^{1-\frac{2-2^{-d+2}}{2^{d-1}(t+1)}+\epsilon }. \end{aligned}$$

(12)

Now (5) and (12) together gives

$$\begin{aligned} D_N(p(n)\alpha )\ll _{\epsilon ,d,t} \frac{1}{H}+N^{-\frac{2-2^{-d+2}}{2^{d-1}(t+1)}+\epsilon }H^{\frac{t}{t+1} +\epsilon }. \end{aligned}$$

Finally we choose $H=\left[ N^{\frac{2-2^{-d+2}}{2^{d-1}(2t+1)}}\right]$ to get

$$\begin{aligned} D_N(p(n)\alpha )\ll _{\epsilon ,d,t} N^{-\frac{2-2^{-d+2}}{2^{d-1}(2t+1)}+\epsilon }. \end{aligned}$$

$\square$

3 Proof of the theorem

Let H be any positive integer which will be chosen later. By Lemma 6, we have

$$\begin{aligned} D_N([p(n)\alpha ]\beta ) \le \frac{2}{H+1}+ \frac{2}{N} \sum _{h=1}^H\frac{1}{h}\left| \sum _{n=0}^{N-1} e(h[p(n)\alpha ]\beta )\right| . \end{aligned}$$

(13)

Recall that $f_{\tau }(x)=e(\tau \{x\})$ and $g_{\tau ,\delta }$ is defined as in (2) with $\delta :=\delta (h)=h^{-1}N^{-\theta }$ for some $0<\theta <1$. Writing $[x]=x-\{x\}$ we have

$$\begin{aligned} \sum _{n=0}^{N-1} e(h[p(n)\alpha ] \beta )&= \sum _{n=0}^{N-1} e(h p(n)\alpha \beta )f_{-h\beta }(p(n) \alpha ) \nonumber \\&= \sum _{n=0}^{N-1} e(h p(n)\alpha \beta )g_{-h\beta ,\delta }(p(n) \alpha ) \nonumber \\&\quad + O\left( \sum _{n=0}^{N-1} | f_{-h\beta }(p(n) \alpha ) - g_{-h\beta ,\delta }(p(n) \alpha )| \right) . \end{aligned}$$

(14)

By Lemma 1 for the O-term on the right hand side of (14) and substituting it in the inequality (13) we have

$$\begin{aligned} D_N([p(n) \alpha ] \beta )\ll & {} \frac{1}{H} + \frac{1}{N}\sum _{h=1}^H \frac{1}{h} \left| \sum _{n=0}^{N-1} e(h p(n)\alpha \beta )g_{-h\beta ,\delta }(p(n) \alpha )\right| \\&+ r \sum _{h=1}^H \frac{\delta }{h} + |\beta | r^2 \sum _{h=1}^H \delta + D_N(p(n) \alpha ) \log H. \end{aligned}$$

The Fourier inversion formula for $g_{\tau ,\delta }$ gives us

$$\begin{aligned} D_N([p(n) \alpha ] \beta )\ll & {} \frac{1}{N}\sum _{h=1}^H \frac{1}{h} \left| \sum _{k\in \mathbb Z} \hat{g}_{-h\beta ,\delta }(k)\sum _{n=0}^{N-1} e(p(n)\alpha (h\beta -k))\right| + \frac{1}{H} \nonumber \\&+\, r \sum _{h=1}^H \frac{\delta }{h} + |\beta | r^2 \sum _{h=1}^H \delta + D_N(p(n) \alpha ) \log H. \end{aligned}$$

(15)

Let

$$\begin{aligned} S_N\; =\; \frac{1}{N}\sum _{h=1}^H \frac{1}{h} \left| \sum _{k\in \mathbb Z} \hat{g}_{-h\beta ,\delta }(k)\sum _{n=0}^{N-1} e(p(n)\alpha (h\beta -k))\right| . \end{aligned}$$

(16)

Let $\rho$ be a real number such that $\rho \in [1,2]$, which will be chosen later. We also suppose $N^{\theta } > 2|\beta |$. Splitting the first sum inside the modulus into $|k|>h^\rho N^{\theta }$ and $|k|\le h^{\rho }N^{\theta }$ gives us

$$\begin{aligned} \begin{aligned} S_N\;&\ll \; \frac{1}{N}\sum _{h=1}^H \frac{1}{h} \left| \sum _{|k|\le h^{\rho }N^{\theta }} \hat{g}_{-h\beta ,\delta }(k)\sum _{n=0}^{N-1} e(p(n)\alpha (h\beta -k))\right| \\&\quad +\sum _{h=1}^H \frac{1}{h} \sum _{|k|> h^\rho N^{\theta }} |\hat{g}_{-h\beta ,\delta }(k)|. \end{aligned} \end{aligned}$$

Lemma 3 with $K=h^{\rho }N^{\theta }$ shows that the second term on the right hand side is $\ll H^{r(1-\rho )}$.

Hence we have

$$\begin{aligned} S_N\; \ll \; \frac{1}{N}\sum _{h=1}^H \frac{1}{h} \left| \sum _{|k|\le h^{\rho }N^{\theta }} \hat{g}_{-h\beta ,\delta }(k)\sum _{n=0}^{N-1} e(p(n)\alpha (h\beta -k))\right| +H^{r(1-\rho )}. \end{aligned}$$

(17)

Using Hölder’s inequality

$$\begin{aligned}&\left| \sum _{|k|\le h^{\rho }N^{\theta }}\right. \left. \hat{g}_{-h\beta ,\delta }(k) \sum _{n=0}^{N-1} e(p(n)\alpha (h\beta -k))\right| \nonumber \\&\quad \ll \left( \sum _{|k|\le h^{\rho }N^{\theta }} |\hat{g}_{-h\beta ,\delta }(k)|^{\frac{2^{d-1}}{2^{d-1}-1}} \right) ^{\frac{2^{d-1}-1}{2^{d-1}}} \left( \sum _{|k|\le h^{\rho }N^{\theta }} \left| \sum _{n=0}^{N-1} e(p(n)\alpha (h\beta -k))\right| ^{2^{d-1}} \right) ^{\frac{1}{2^{d-1}}}\nonumber \\&\quad \ll \left( \sum _{|k|\le h^{\rho }N^{\theta }} \left| \sum _{n=0}^{N-1} e(p(n)\alpha (h\beta -k))\right| ^{2^{d-1}} \right) ^{\frac{1}{2^{d-1}}}. \end{aligned}$$

(18)

Here we have used Lemma 4 to get the last inequality.

Let $\xi = \alpha (h\beta -k)$. Using Lemma 5, with $k=d-1$ and $\lambda _m=e(p(m)\xi )$ we get that the following inequalities hold for any $Q\in [1,N]$:

$$\begin{aligned}&\left| \sum _{n=0}^{N-1} e(p(n)\xi )\right| ^{2^{d-1}} \\&\quad \ll \frac{N^{2^{d-1}}}{Q}+\frac{N^{2^{d-1}-1}}{Q^{2-2^{-d+2}}}\sum _{r_1=1}^Q\sum _{ r_2=1}^{Q^{\frac{1}{2}}}\cdots \sum _{r_{d-1}=1}^{Q^{2^{-d+2}}}\left| \sum _{n=0}^{ N-1-r_1-\cdots -r_{d-1}}e(d!r_1\cdots r_{d-1}n\xi )\right| \\&\quad \ll \frac{N^{2^{d-1}}}{Q}+\frac{N^{2^{d-1}-1}}{Q^{2-2^{-d+2}}}\sum _{r_1=1}^Q\sum _{ r_2=1}^{Q^{\frac{1}{2}}}\cdots \sum _{r_{d-1}=1}^{Q^{2^{-d+2}}}\left| \min \left( N, \frac{1}{\Vert d!r_1\cdots r_{d-1}\xi \Vert }\right) \right| , \end{aligned}$$

where we have used $\sum _{n=0}^{N-1}e(n\lambda )\ll \min (N, \frac{1}{\Vert \lambda \Vert })$ to get the last inequality.

Let $T(m)=|\{(r_1,\ldots ,r_{d-1})\in [1,Q]\times \cdots \times [1,Q^{2^{-d+2}}]: r_1\cdots r_{d-1}=m\}|$. With this notation the above inequality will be

$$\begin{aligned} \left| \sum _{n=0}^{N-1} e(p(n)\xi )\right| ^{2^{d-1}}\ll \frac{N^{2^{d-1}}}{Q}+\frac{N^{2^{d-1}-1}}{Q^{2-2^ {-d+2}}}\sum _{m=1}^{Q^{2-2^{-d+2}}}T(m)\min \left( N,\frac{1}{\Vert d!\xi m\Vert }\right) . \end{aligned}$$

Let $\epsilon >0$ be any real number. Let $\epsilon _2=\frac{\epsilon }{(2-2^{-d+2})}$. Since $T(m)\le \tau _{d-1}(m)\ll _{\epsilon _2} m^{\epsilon _2}$, we get

$$\begin{aligned} \left| \sum _{n=0}^{N-1} e(p(n)\xi )\right| ^{2^{d-1}}\ll _{\epsilon ,d} \frac{N^{2^{d-1}}}{Q}+\frac{N^{2^{d-1}-1}}{Q^{2-2^ {-d+2}-\epsilon }}\sum _{m=1}^{Q^{2-2^{-d+2}}}\min \left( N,\frac{1}{\Vert d!\xi m\Vert }\right) .\; \end{aligned}$$

(19)

Now we prove the following lemma which will be used to estimate the right hand side of the above equation.

Lemma 8

Let $\xi =\alpha (h\beta -k)$. Then for any $\epsilon >0$ we have

$$\begin{aligned} \sum _{\ell =1}^L \min \left( N, \frac{1}{\Vert d!\ell \xi \Vert } \right) \ll _{\alpha ,\beta ,\epsilon ,d} L\log N +NL^{1-\frac{1}{2t+1}+\epsilon }(h|k|)^{\frac{t}{2t+1}+\epsilon }\log N. \end{aligned}$$

Proof

For $0\le m \le N-1$, define

$$\begin{aligned} E_m=\left\{ \ell \le L: \frac{m}{N}<\Vert d!\ell \xi \Vert \le \frac{m+1}{N} \right\} . \end{aligned}$$

We have

$$\begin{aligned} \sum _{\ell =1}^L \min \left( N, \frac{1}{\Vert d!\ell \xi \Vert } \right)= & {} N |E_0| + \sum _{l \notin E_0} \frac{1}{\Vert d!\ell \xi \Vert } \\\le & {} N |E_0| + \sum _{m=1}^{N-1} \frac{N}{m} |E_m|. \end{aligned}$$

Observe that

$$\begin{aligned} |E_k| = \frac{2L}{N} + O( L D_L(d!\ell \xi ) ). \end{aligned}$$

Thus

$$\begin{aligned} \sum _{\ell =1}^L \min \left( N, \frac{1}{\Vert d!\ell \xi \Vert } \right) \ll L \log N + NL D_L(d!\ell \xi )\log N . \end{aligned}$$

(20)

Using Lemma 7 and the fact that

$$\begin{aligned} \Vert d!\ell \xi \Vert = \Vert d!\ell \alpha (h\beta -k)\Vert \ge \frac{C(\alpha ,\beta ,\epsilon )}{((d!\ell )^2 h |k| )^{t+\epsilon } } \end{aligned}$$

for any positive integer $\ell \ge 1$, we get

$$\begin{aligned} D_L(d!\ell \xi ) \ll _{\alpha ,\beta ,\epsilon } \frac{1}{m}+ \frac{1}{L}(h|k|(d!m)^2)^{t+\epsilon } \end{aligned}$$

for any positive integer m. Now we choose $m=L^{1/(2t+1)}(h|k|)^{-t/(2t+1)}$ to get

$$\begin{aligned} D_L(d!\ell \xi ) \ll _{\alpha ,\beta ,\epsilon ,d} (h^t |k|^t)^{\frac{1}{2t+1}+\epsilon } L^{-\frac{1}{2t+1}+\epsilon } . \end{aligned}$$

(21)

Substituting the above estimate in (20) gives us

$$\begin{aligned} \sum _{\ell =1}^L \min \left( N, \frac{1}{\Vert d!\ell \xi \Vert } \right) \ll _{\alpha ,\beta ,\epsilon ,d} L \log N + NL ^{1-\frac{1}{2t+1}+\epsilon } (h|k|)^{\frac{t}{2t+1}+\epsilon }\log N . \end{aligned}$$

$\square$

Apply Lemma 8 in (19) with $L=Q^{2-2^{-d+2}}$ and let $Q=N$ to get

$$\begin{aligned} \left| \sum _{n=0}^{N-1} e(p(n)\xi )\right| ^{2^{d-1}}\ll _{\alpha ,\beta ,\epsilon ,d} N^{2^{d-1}-1}+N^{2^{d-1}-\left( \frac{2-2^{-d+2}}{2t+1}\right) +\epsilon }h^{ \frac{t}{2t+1}+\epsilon }|k|^{\frac{t}{2t+1}+\epsilon }. \end{aligned}$$

(22)

Summing both sides of the above inequality over k we get that

$$\begin{aligned} \begin{aligned} \sum _{|k|\le h^{\rho }N^{\theta }}\left| \sum _{n=0}^{N-1} e(p(n)\xi )\right| ^{2^{d-1}}&\ll _{\alpha ,\beta ,\epsilon ,d} N^{2^{d-1}-1+\theta }h^{\rho } \\&\quad +N^{2^{d-1}-(\frac{2-2^{-d+2}}{2t+1})+\theta (\frac{ 3t+1}{2t+1})+\epsilon }h^{\frac{t}{2t+1}+\rho (\frac{3t+1}{2t+1})+\epsilon }. \end{aligned} \end{aligned}$$

(23)

Clearly the first term on the right hand side is dominated by the second term. Putting this inequality in (18) we get that

$$\begin{aligned}&\left| \sum _{|k|\le h^{\rho }N^{\theta }} \hat{g}_{-h\beta }(k) \right. \left. \sum _{n=0}^{N-1} e(p(n)\alpha (h\beta -k))\right| \nonumber \\&\quad \ll _{\alpha ,\beta ,\epsilon ,d} N^{1-(\frac{2-2^{-d+2}}{2^{d-1}(2t+1)})+\theta (\frac{3t+1}{2^{d-1}(2t+1)} )+\epsilon }h^{\frac{t}{2^{d-1}(2t+1)}+\rho (\frac{3t+1}{2^{d-1}(2t+1)} )+\epsilon }. \end{aligned}$$

Hence we have

$$\begin{aligned}&\sum _{h=1}^H\frac{1}{h}\left| \sum _{|k|\le h^{\rho }N^{\theta }} \hat{g}_{-h\beta }(k) \right. \left. \sum _{n=0}^{N-1} e(p(n)\alpha (h\beta -k))\right| \nonumber \\&\quad \ll _{\alpha ,\beta ,\epsilon ,d} N^{1-(\frac{2-2^{-d+2}}{2^{d-1}(2t+1)})+\theta (\frac{3t+1}{2^{d-1}(2t+1)} )+\epsilon }H^{\frac{t}{2^{d-1}(2t+1)}+\rho (\frac{3t+1}{2^{d-1}(2t+1)} )+\epsilon }. \end{aligned}$$

From (17) and above inequality we have

$$\begin{aligned} S_N\ll _{\alpha ,\beta ,\epsilon ,d} N^{-(\frac{2-2^{-d+2}}{2^{d-1}(2t+1)})+\theta (\frac{3t+1}{2^{d-1}(2t+1)} )+\epsilon }H^{\frac{t}{2^{d-1}(2t+1)}+\rho (\frac{3t+1}{2^{d-1}(2t+1)} )+\epsilon }+O(H^{r(1-\rho )}). \end{aligned}$$

Hence we have from (15) with $\delta ^{-1}=hN^{\theta }$ that

$$\begin{aligned}&D_N([p(n)\alpha ]\beta ) \\&\quad \ll _{\alpha ,\beta ,\epsilon ,d} H^{\frac{t}{2^{d-1}(2t+1)}+\rho (\frac{3t+1}{2^{d-1}(2t+1)})+\epsilon } N^{-(\frac{2-2^{-d+2}}{2^{d-1}(2t+1)})+\theta (\frac{3t+1}{2^{d-1}(2t+1)} )+\epsilon } +H^{r(1-\rho )}\\&\qquad +\frac{1}{H}+N^{-\theta }+N^{-\theta }\log H +D_N(p(n)\alpha )\log H\;. \end{aligned}$$

We choose $\rho =1+\epsilon _1$ with $\epsilon _1=\epsilon _1(\epsilon ,t)>0$ sufficiently small real number, and r is an integer satisfying $r>\frac{1}{\epsilon _1}$. Hence the second term on the right hand side is $\ll H^{-1}$.

Now we choose $H=[N^{\theta }]$ with $\theta =\frac{2-2^{-d+2}}{2^{d-1}(2t+1)+(4t+1)+\rho (3t+1)}$. With these choices we have

$$\begin{aligned} D_N([p(n)\alpha ]\beta ) \ll _{\alpha ,\beta ,\epsilon ,d} N^{-\frac{2-2^{-d+2}}{2^{d-1}(2t+1)+7t+2}+\epsilon }+D_N(p(n)\alpha )\log N. \end{aligned}$$

(24)

By Lemma 1, we have

$$\begin{aligned} D_N(p(n)\alpha )\ll _{\epsilon ,d,t} N^{-\frac{2-2^{-d+2}}{2^{d-1}(2t+1)}+\epsilon }. \end{aligned}$$

Putting this in (24), we get

$$\begin{aligned} D_N([p(n)\alpha ]\beta ) \ll _{\alpha ,\beta ,\epsilon ,d} N^{-\frac{2-2^{-d+2}}{2^{d-1}(2t+1)+7t+2}+\epsilon }. \end{aligned}$$