Kloosterman sums with twice-differentiable functions

TL;DR

This paper establishes bounds on Kloosterman-like sums involving twice-differentiable functions with specific curvature conditions, leading to new insights into the distribution of modular inverses of sequences like Piatetski-Shapiro sequences.

Contribution

It introduces novel bounds for sums involving the floor of twice-differentiable functions and applies these to analyze the distribution of modular inverses in sequences such as Piatetski-Shapiro sequences.

Findings

Bounds on Kloosterman-like sums for specific functions

Results on the distribution of modular inverses in certain sequences

Application to Piatetski-Shapiro sequences with c in (1, 4/3)

Abstract

We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2\pi i (x \lfloor f(n)\rfloor+ y \lfloor f(n)\rfloor^{-1})/p), \] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f''$, and $\lfloor f(n)\rfloor^{-1}$ is meaning inversion modulo~$p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $\lfloor f(n)\rfloor^{-1} \pmod{p}$. The results apply, in particular, to Piatetski-Shapiro sequences $ \lfloor t^c\rfloor$ with $c\in(1,\frac{4}{3})$. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from 2006 to 2009.

Equations

$$\sum _{n=1}^N\exp (2\pi i(x\lfloor f(n)\rfloor +y\lfloor f(n){\rfloor }^{-1})/p),$$

$$\#\{n\le N:\text{leavevmode}\lfloor n^c\rfloor \text{ prime}\}=(1+o(1))\frac{N}{c\log N}\text{as }N\to \infty$$

$$\sum _{n=1}^N\chi (\lfloor f(n)\rfloor )(N\ge 1),$$

$${\psi }_{x,y;p}(u)=\{\begin{array}{ll}{e}_p(xu+y{u}^{-1})& \text{if }u\ne 0,\\ 0& \text{otherwise},\end{array}$$

$$\underset{t\to \infty }{\lim }\frac{\log f^{\prime \prime }(t)}{\log t}=-\kappa .$$

$$p^{1/(2\kappa )+\epsilon }\le N<p^{1/(2-\kappa )},$$

$$\sum _{n=1}^N{\psi }_{x,y;p}(\lfloor f(n)\rfloor ){\ll }_{\epsilon ,f}N{p}^{-\delta }$$

$$\delta =2^{-11}{\epsilon }^2{\kappa }^4.$$

$$T_f(N,\mathcal{I})=\left\{n:\text{leavevmode}\lfloor f(n){\rfloor }^{-1}\equiv h(\mathrm{mod}p),1\le n\le N,h\in \mathcal{I}\right\}.$$

$$T_f(N,\mathcal{I})=\frac{HN}{p}+O_{\epsilon ,f}(N{p}^{-\delta }\log p).$$

$$p\ge H,N\text{mbox}andHN\ge p^{2-\xi }$$

$$\sum _{n=1}^N{\psi }_{x,y;p}(\lfloor f(n)\rfloor )$$

$$N^{1-\kappa /2}\ge p^{3c},N^{\kappa /2-c}\ge p^{3c},N\ge p^{1/2+3c},$$

$$N^{\kappa -c}\ge p^{1/2+3c}.$$

$$K_{j-1}-K_j=\Delta K_j,j=1,\dots ,R.$$

$$R=\lfloor N^{1+c-\kappa }{\log }^{2}p\rfloor \text{mbox}and\Delta =p^{c/R}-1,$$

$$K_j=p^{-jc/R}N,j=0,\dots ,R.$$

$$K^{\kappa -\epsilon }\ge L\ge K^{\kappa /2}{p}^{\epsilon },K\le p^{1/(2-\kappa )},L\ge p^{1/2+\epsilon },$$

$$\sum _{K<n\le K+L}{\psi }_{x,y;p}(\lfloor f(n)\rfloor ){\ll }_{\epsilon ,f}L{p}^{-{\delta }_0}$$

$$\sum _{K<n\le K+L}{\psi }_{x,y;p}(\lfloor f(n)\rfloor )=\sum _{K<n\le K+L}{\psi }_{x,y;p}(\lfloor f(n+h)\rfloor )+O(h).$$

$$\sum _{K<n\le K+L}{\psi }_{x,y;p}(\lfloor f(n)\rfloor )=\frac{1}{H}\sum _{h=1}^H\sum _{K<n\le K+L}{\psi }_{x,y;p}(\lfloor f(n+h)\rfloor )+O(H).$$

$$f(n+h)=f(n)+h{f}^{\prime }(K)+I_{n,h}+J_{n,h},$$

$$I_{n,h}=h{\int }_K^n{f}^{\prime \prime }(u)\mathrm{du}\text{mbox}and{J}_{n,h}={\int }_n^{h+n}{f}^{\prime \prime }(u)(h+n-u)\mathrm{du}.$$

$$0\le I_{n,h}+J_{n,h}<1.$$

$$\lfloor f(n+h)\rfloor =\lfloor f(n)+{\xi }_0\rfloor +\lfloor h{f}^{\prime }(K)-{\xi }_0\rfloor +{\eta }_{n,h,{\xi }_0},$$

$$f(n)+{\xi }_0,h{f}^{\prime }(K)-{\xi }_0,I_{n,h}{J}_{n,h}.$$

$$\sum _{u\in \mathcal{U}}\sum _{v\in \mathcal{V}}{a}_u{b}_v{\psi }_{x,y;p}(u+v),$$

$$\sum _{u\in \mathcal{U}}\sum _{v\in \mathcal{V}}{a}_u{b}_v{\psi }_{x,y;p}(u+v){\ll }_{k}AB{U}^{1-1/(2k)}(V^{1/2}{p}^{1/(2k)}+V{p}^{1/(4k)})$$

$$A=\underset{u\in \mathcal{U}}{\max }|a_u|\text{mbox}andB=\underset{v\in \mathcal{V}}{\max }|b_v|.$$

$${\mathfrak{S}}^{2k}$$

Full text

[TABLE]

with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f^{\prime\prime}$, and $\lfloor f(n)\rfloor^{-1}$ is meaning inversion modulo $p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $\lfloor f(n)\rfloor^{-1}\pmod{p}$. The results apply, in particular, to Piatetski-Shapiro sequences $\lfloor t^{c}\rfloor$ with $c\in(1,\frac{4}{3})$. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from 2006 to 2009.

1. Introduction and main result

1.1. Background and motivation

The Piatetski-Shapiro sequence associated with $c\in(1,2)$ is defined by $(\lfloor n^{c}\rfloor)_{n\in\mathbb{N}}$, where $\lfloor x\rfloor=\min\{m\in\mathbb{Z}:m\leq x\}$ denotes the floor function. Such sequences are named in honour of Pyateckiĭ-Šapiro [25]111Nowadays his name is usually spelled as Piatetski-Shapiro. who, at the suggestion of A. O. Gelfond, has proved the following prime number theorem:

[TABLE]

for $c$ in the range $1<c<\frac{12}{11}$. Such a result may be viewed as an intermediate step to tackling the problem of investigating the number of primes represented by a fixed quadratic polynomial; consider for instance Landau’s famous problem of ascertaining whether $n^{2}+1$ is prime for infinitely many $n\in\mathbb{N}$. Informally speaking, the upper bound for the exponents $c$ for which one is able to establish Eq. 1.1\wrtusdrfeq:Piatetski-Shapiro:PNT measures the progress towards quadratic polynomials. To date, the largest admissible $c$-range seems to be $1<c<\frac{2817}{2426}$ due to Rivat and Sargos [27] (see also the references to the previous record holders they give in their paper). Naturally, also lower bound sieves have been employed, and the corresponding current record is a version of Eq. 1.1\wrtusdrfeq:Piatetski-Shapiro:PNT with a lower bound of the right order of magnitude instead of an asymptotic formula and $1<c<\frac{243}{205}$ due to Rivat and Jie [26].

Investigations into arithmetic properties of Piatetski-Shapiro sequences are not confined to studying prime values; they have been studied with respect to various other topics, including, but not limited to, the following:

• •

smooth, rough, and square-free numbers [6, 2, 1],

• •

almost-primes [7, 9],

• •

additive problems [22, 4, 20],

• •

intersection with special sequences [13, 8, 5, 3, 19], and

• •

digital expansions [21, 28, 24].

For a broader picture of the scope of each topic, we refer the interested reader to the references within the cited items.

Here we study a question of distribution of modular inverses modulo a prime $p$ of Piatetski-Shapiro sequences and, in fact, more general sequences. Our motivation comes from [29], where the distribution of inverses modulo a prime $p$ of Beatty sequences is considered. Furthermore, since this question immediately leads us to a problem of estimating Kloosterman-like sums with Piatetski-Shapiro sequences, this serves as an additional motivation. Indeed, this is an additive analogue of the results from [13], which concern bounds for character sums of the form

[TABLE]

where $f$ is a real-valued, twice-differentiable function satisfying a certain limit condition on $f^{\prime\prime}$ (see Eq. 1.4\wrtusdrfeq:2ndDerivativeGrowth below), $\chi$ is a non-trivial multiplicative character modulo a prime $p$, which is assumed to be in a suitable range with respect to $N$. The results from [13] apply, in particular, to power functions $f(t)=t^{c}$ with $c\in(1,\frac{4}{3})$ and apply, amongst other things, to bounding the least quadratic non-residue in Piatetski-Shapiro sequences (see also [8, 5]).

1.2. Notation

Throughout the paper, the notation $U=O(V)$, $U\ll V$ and $V\gg U$ are equivalent to $|U|\leq c|U|$ for some positive constant $c$, which, may occasionally depend on the function $f$ and on the small positive real parameter $\varepsilon$ and on the positive integer parameter $k$. We use subscripts to indicate such dependencies.

We always use $p$ to denote a prime number and then for $x,y,u\in\mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}$, we define

[TABLE]

where $\operatorname{e}_{p}\lparen a\rparen=\exp(2\pi ia/p)$ and the calculation inside the argument of $\operatorname{e}_{p}\lparen\,\cdot\,\rparen$ is to be performed in $\mathbb{F}_{p}$.

1.3. Main results

In this paper, we first outline how to adapt the arguments from [13] to bound exponential sums with $\psi_{x,y;\,p}(\lfloor f(n)\rfloor)$ which immediately provide some non-trivial information about the distribution of modular inverses $\lfloor f(n)\rfloor^{-1}\pmod{p}$. We remark that for the scenario of [10, 11, 12], any non-trivial bound on the character sum implies the desired result about the distribution of quadratic non-residues. We also use the opportunity to provide slightly more precise information about the dependence of our saving $\delta$ on the parameter $\kappa$, characterising the growth of $f$. In particular, the explicit formula Eq. 1.7\wrtusdrfeq:delta below shows that $\delta$ is a monotonic function of $\varepsilon$ and $\kappa$.

Our main result may be stated as follows:

Theorem 1.1**.**

Let $\frac{2}{3}<\kappa<1$, and let $f$ be a real-valued, twice-differentiable function such that

[TABLE]

Then, for a sufficiently small $\varepsilon>0$, for all $\psi_{x,y;\,p}$, with $p$ prime and $x,y,u\in\mathbb{F}_{p}$, as in Eq. 1.3\wrtusdrfeq:psi:definition and all integers $N$ in the range

[TABLE]

the uniform bound

[TABLE]

holds with

[TABLE]

For an interval $\mathscr{I}=[K+1,K+H-1]\subseteq[0,p-1]$ with integers $K$ and $H$, we denote by $T_{f}(N,\mathscr{I})$ the number of positive integers $n\leq N$ for which the smallest positive residue $\lfloor f(n)\rfloor^{-1}\pmod{p}$ falls in $\mathscr{I}$, that is

[TABLE]

Corollary 1.2**.**

On the hypotheses of 1.1\wrtusdrfthm:main:result, we have

[TABLE]

where $\delta$ is given by Eq. 1.7\wrtusdrfeq:delta.

Corollary 1.3**.**

Let $f$ be a real-valued, twice-differentiable function such that Eq. 1.4\wrtusdrfeq:2ndDerivativeGrowth holds. There exists a constant $\xi>0$ which depends only on $\kappa$, such that for

[TABLE]

we have $T_{f}(N,\mathscr{I})>0$.

Note that 1.3\wrtusdrfcor:Exist is an analogue of [29, Theorem 5.1], where a result of this type is given for Beatty sequences.

2. Outline of the argument

2.1. Preliminaries

As large parts of the arguments in [13] essentially carry over verbatim to the proof of 1.1\wrtusdrfthm:main:result, we choose not to repeat them here in full detail. Instead, we give an informal description of the underlying argument. The argument ultimately relies on a bound for certain double sums, [13, Lemma 4.1], and it is this bound which we need to adapt to our setting, A proof of this adapted bound, which is contained in our 3.2\wrtusdrfcor:DoubleSumBound-eps below, is then carried out in full detail in Section 3.1\wrtusdrfsec:DoubleSumBound. Some further details pertaining to the explicit value of $\delta$ given in 1.1\wrtusdrfthm:main:result are contained in Sections 3.2 and 3.3\wrtusdrfsec:ExplicitLemma,sec:main proof.

2.2. Reduction from long sums to sums over short intervals

To get started, consider the sum

[TABLE]

from 1.1\wrtusdrfthm:main:result. We now fix some constant $c>0$ (whose final choice depends on $\kappa$ and $\varepsilon$), to satisfy the inequalities

[TABLE]

as well as

[TABLE]

Next, for some parameter $R$, the sum (2.1) can be decomposed into $R$ sum over a small initial segment, $n\leq Np^{-c}$ and $R$ ‘short’ sums over $K_{j}<n\leq K_{j-1}$ with $j=1,\ldots,R$ and numbers $N=K_{0}>K_{1}>\ldots>K_{R}=Np^{-c}$ satisfying

[TABLE]

for some parameter $\Delta$.

Concerning the sum over the initial segment, already the trivial estimate is satisfactory, as it is within the bound Eq. 1.6\wrtusdrfeq:main:result of 1.1\wrtusdrfthm:main:result. The remaining short sums are then treated with 2.1\wrtusdrflem:ShortSumBound below. Of course, to do this in the first place, the parameters $R$ and $\Delta$ have to be chosen appropriately. The specific choice used in [13], which also works in the setting of this paper, is (with the above choice of $c$)

[TABLE]

and then

[TABLE]

One now verifies that Eq. 2.4\wrtusdrfeq:short int holds. We plainly refer to [13] for the technical details.

2.3. Reduction from sums over short intervals to double sums

For the short sums, one can use the following result: (This is already adapted to our setting; see [13, Lemma 5.1] for the corresponding character sum variant.)

Lemma 2.1**.**

Fix $\varepsilon>0$ and $\frac{2}{3}<\kappa<1$. Let $f$ be a real-valued, twice-differentiable function satisfying Eq. 1.4\wrtusdrfeq:2ndDerivativeGrowth. Then, for all $\psi_{x,y;\,p}$, with $p$ prime and $x,y,u\in\mathbb{F}_{p}$, as in Eq. 1.3\wrtusdrfeq:psi:definition, and all real numbers $K,L$ that satisfy the inequalities

[TABLE]

the uniform bound

[TABLE]

holds with some constant $\delta_{0}>0$ that may only depend on $\varepsilon$ and $\kappa$.

Next, we sketch the idea of the proof of 2.1\wrtusdrflem:ShortSumBound. Trivially, for every integer $h\geq 0$, one has

[TABLE]

Therefore, upon averaging over all $h\leq H$ for some parameter $H$,

[TABLE]

To progress further, one would like to separate the variables $n$ and $h$ in the argument of $\psi_{x,y;\,p}$. This is achieved using the following formula:

[TABLE]

where

[TABLE]

In view of Eq. 1.4\wrtusdrfeq:2ndDerivativeGrowth, the relevant integrals can be seen to be acceptably small provided that $p$ is large enough. In particular, the last observation also crucially relies upon $n$ being not too large with respect to $K$, which is the case, because we are dealing with short sums. We now suppose that

[TABLE]

However, there still is a complication with this approach arising through the floor function. Indeed, for any $\xi_{0}\in[0,1\rparen$,

[TABLE]

with some undesired correction term $\eta_{n,h}\in\{0,1,2\}$ depending on both $n$ and $h$ (and $\xi_{0}$). To get around this, one restricts the averaging to only those $h\leq H$ for which the fractional part of $hf^{\prime}(K)-\xi_{0}$ is small. By a clever choice of $\xi_{0}$, using the pigeonhole principle, one can ensure that the set of $h\leq H$ having the desired property is not too sparse without actually having to know anything about the distribution of the fractional parts of $hf^{\prime}(K)$ as $h=1,2,\ldots,H$. Then, in Eq. 2.6\wrtusdrfeq:AveragedSum one restricts to only those $n$ such that the fractional part of $f(n)+\xi_{0}$ is bounded away from $1$ in such a way that no carry to a next integer occurs when adding the four terms

[TABLE]

Clearly, these choices of $n$ and $h$ ensure that Eq. 2.7\wrtusdrfeq:carry holds with $\eta_{n,h,\xi_{0}}=0$. Finally, one can see that the number of $n$ which had to be discarded from Eq. 2.6\wrtusdrfeq:AveragedSum is not too large. In [13] this is accomplished by bounding the discrepancy of the sequence of fractional parts of $f(n)$ as $n=1,2,\ldots,N$, using the Erdős–Turán inequality to translate this to a problem of estimating certain exponential sums, and estimating these sums using a standard application of the van der Corput method.

The above argument, and in particular the additive split Eq. 2.7\wrtusdrfeq:carry, reduces the proof of 2.1\wrtusdrflem:ShortSumBound to bounding double sums of the shape

[TABLE]

where $\mathscr{U},\mathscr{V}$ are subsets of $\mathbb{F}_{p}$ and $a_{u},b_{v}$ ($u\in\mathscr{U},v\in\mathscr{V}$) are certain weights. We also recall Eq. 2.6\wrtusdrfeq:AveragedSum and the subsequent discussion about separating $n$ and $h$; the need for including the weights arises from the potential failure of, e.g., $\lfloor f(n)+\xi_{0}\rfloor$ to produce only distinct values modulo $p$ as $n$ varies.

2.4. Concluding the proof

Given the above discussion, it is evident that, up to carrying out the technicalities which are, however, all readily found in [13], 1.1\wrtusdrfthm:main:result follows from 2.1\wrtusdrflem:ShortSumBound, and in turn 2.1\wrtusdrflem:ShortSumBound may be deduced from appropriate bounds for double sums Eq. 2.8\wrtusdrfeq:double sum. Such a suitable bound is given in 3.2\wrtusdrfcor:DoubleSumBound-eps below.

3. Technical Details

3.1. Bounds for certain double sums

Here we prove a bound on the double sums Eq. 2.8\wrtusdrfeq:double sum which concludes the proof of 1.1\wrtusdrfthm:main:result. In fact, we first give a slightly more general bound:

Lemma 3.1**.**

Suppose that $p$ is prime and $\mathscr{U},\mathscr{V}$ are subsets of $\mathbb{F}_{p}$ of cardinalities $U=\#\mathscr{U}$ and $V=\#\mathscr{V}$. Then, for an arbitrary fixed integer $k$, for any complex numbers $a_{u},b_{v}$ ($u\in\mathscr{U},v\in\mathscr{V}$) and $x,y\in\mathbb{F}_{p}$ with $y\neq 0$, we have

[TABLE]

where

[TABLE]

Proof.

Denote the left hand side of Eq. 3.1\wrtusdrfeq:DoubleSumBound by $\mathfrak{S}$. Then we apply Hölder’s inequality and subsequently extend the summation over $u\in\mathscr{U}$ to $u\in\mathbb{F}_{p}$, getting

[TABLE]

Therefore,

[TABLE]

where

[TABLE]

and $\smash{\sum^{*}}$ restricts the summation to those $u\in\mathbb{F}_{p}$ such that all of the numbers

[TABLE]

that is, are non-zero in $\mathbb{F}_{p}$. Thus,

[TABLE]

We can assume that $p>k$ as otherwise there is nothing to prove. Examining the poles of the rational function $R_{\bm{v},\bm{w}}(X)$, we see that it is constant, and in fact vanishes, only if the vectors $\bm{v}$ and $\bm{w}$ differ only by a permutation of their components. This happens only for $O_{k}(V^{k})$ choices of $\bm{v},\bm{w}\in\mathscr{V}^{k}$. For such choices we estimate the inner most sum in Eq. 3.2\wrtusdrfeq:UseBombieriWeilHere trivially as $p$. Hence the total contribution to Eq. 3.2\wrtusdrfeq:UseBombieriWeilHere from such $\bm{v},\bm{w}\in\mathscr{V}^{k}$ is $O_{k}(V^{k}p)$.

For the remaining $O(V^{2k})$ choices we use the Weil bound of exponential sums with rational functions (see, for example, [23, Theorem 2], several more general bounds can also be found in [15]) and conclude that the total contribution to Eq. 3.2\wrtusdrfeq:UseBombieriWeilHere from such $\bm{v},\bm{w}\in\mathscr{V}^{k}$ is $O_{k}(V^{2k}p^{1/2})$.

Hence,

[TABLE]

and the result follows. ∎

Corollary 3.2**.**

For any $\varepsilon>0$, in the setting of 3.1\wrtusdrflem:DoubleSumBound, and assuming that

[TABLE]

we have

[TABLE]

Proof.

Taking $k=\lceil 1/(2\varepsilon)\rceil$ in 3.1\wrtusdrflem:DoubleSumBound we have $V\geq p^{1/(2k)}$ and thus the right hand side of Eq. 3.1\wrtusdrfeq:DoubleSumBound can be replaced with

[TABLE]

Since $U\leq p$, we have $\varepsilon\leq 1/2$ and thus one verifies that $2k\leq\varepsilon^{-1}$. The result now follows. ∎

3.2. Explicit version of 2.1\wrtusdrflem:ShortSumBound

Before being able to address the explicit choice of $\delta$ given in Eq. 1.7\wrtusdrfeq:delta, we need an explicit version of 2.1\wrtusdrflem:ShortSumBound. We remark that in the next result the saving $\delta_{0}$ is independent of $\kappa$.

Lemma 3.3**.**

In the setting of 2.1\wrtusdrflem:ShortSumBound, assuming $\varepsilon$ to be sufficiently small, one may take $\delta_{0}=\varepsilon^{2}/26$.

Proof.

We extract only the relevant part of the proof of [13, Theorem 5.1] (adjusted to our setting): we have

[TABLE]

where

• •

$\delta_{1}>0$ is from [13, Equation (12)]

• •

${\mathscr{B}}$ is the bound obtained from 3.2\wrtusdrfcor:DoubleSumBound-eps applied with

[TABLE]

and weights $a_{u},b_{u}$ of size $p^{o_{f}(1)}$ as $p\to\infty$.

Therefore, by 3.2\wrtusdrfcor:DoubleSumBound-eps and Eq. 2.5\wrtusdrfeq:ShortSumBound:Ranges,

[TABLE]

Now only the value of $\delta_{1}$ needs closer inspection. Looking at the lines that precede [13, Equation (12)], and recalling Eq. 2.5\wrtusdrfeq:ShortSumBound:Ranges, one may check that, for $\varepsilon$ sufficiently small, $\delta_{1}=\varepsilon^{2}$ is admissible. Upon plugging this in the above bound, the result follows. ∎

3.3. Proof of 1.1\wrtusdrfthm:main:result

Assume that $\varepsilon>0$ is sufficiently small. Then the inequalities Eq. 2.2\wrtusdrfeq:N:and:p:relations:i_to_iii are all implied by Eq. 2.3\wrtusdrfeq:N:and:p:relations:iv, and the latter is clearly satisfied when choosing

[TABLE]

A close inspection of the proof of [13, Theorem 6.1] shows that

[TABLE]

where $\delta_{1}$ is now any admissible exponent $\delta$ of $p$ from the bound obtained from 2.1\wrtusdrflem:ShortSumBound with $\varepsilon$ replaced by $c$. In particular, by 3.3\wrtusdrflem:ShortSumBound:regularity, we may choose $\delta_{1}=c^{2}/26$. This shows that in Eq. 1.6\wrtusdrfeq:main:result, again assuming $\varepsilon$ to be sufficiently small, we may take $\delta$ as in Eq. 1.7\wrtusdrfeq:delta, which concludes the proof of 1.1\wrtusdrfthm:main:result.

3.4. Proofs of 1.2\wrtusdrfcor:DistrInt and 1.3\wrtusdrfcor:Exist

1.2\wrtusdrfcor:DistrInt follows at once if one combines 1.1\wrtusdrfthm:main:result with the Erdős–Turán inequality (see, for instance, [17, Theorem 1.21]).

To prove 1.3\wrtusdrfcor:Exist, we note that $HN>p^{2-\xi}$ implies that $H,N>p^{1-\xi}$. In particular, assuming $\xi<1/4$ we see that $N$ satisfies the necessary lower bound in Eq. 1.5\wrtusdrfeq:ineq N. We now define

[TABLE]

thus $T_{f}(N,\mathscr{I})\geq T_{f}(N_{0},\mathscr{I})$ and we also see that 1.2\wrtusdrfcor:DistrInt applies to $T_{f}(N_{0},\mathscr{I})$. Taking

[TABLE]

we see that we can assume that $N_{0}=\lceil p^{1/(2-\kappa)}\rceil-1$. Since $H>p^{1-\xi}$ we have the desired result.

4. Comments

4.1. Some predecessors of our approach

Results of the shape of 3.2\wrtusdrfcor:DoubleSumBound-eps in the case Dirichlet characters appear to have been developed by Karatsuba [18] building on earlier work of Davenport and Erdős [16, Lemma 3] and Burgess [14, Lemma 2]. Indeed, the proof of our 3.2\wrtusdrfcor:DoubleSumBound-eps proceeds in a similar vein and the averaging procedure underlying Eq. 2.6\wrtusdrfeq:AveragedSum has been used extensively (again, see [18] and the references therein).

The method has been subsequently adapted in a series of works [10, 11, 12] on properties of Beatty sequences, which often amounts to studying sums of the shape Eq. 1.2\wrtusdrfeq:CharacterSum with $f(t)=\alpha t+\beta$ (with an irrational $\alpha>1$ and real $\beta$) and $\chi$ potentially replaced with some other function of arithmetic interest. Here, in contrast to the situation in Eq. 1.4\wrtusdrfeq:2ndDerivativeGrowth which opens up the possibility of using the van der Corput method, the quality of error terms generally also depends on the Diophantine properties of $\alpha$.

4.2. Further problems

We have not made any attempt at optimising the value of $\delta$ for which one can prove 1.1\wrtusdrfthm:main:result. It would be interesting to see how large a value of $\delta$ the method from [13] can produce if all parameters are chosen optimally. Likewise, ascertaining the sharpest form of 1.3\wrtusdrfcor:Exist would also be interesting.

Moreover, any improvement on 3.1\wrtusdrflem:DoubleSumBound, even for narrower ranges of $U$ and $V$, would be of independent interest. As a first step in this direction we record the following result which is non-trivial whenever $UV\gg p$ and, in this range and up to the implied constants, at least as good as 3.1\wrtusdrflem:DoubleSumBound and stronger when $U$ and $V$ are both large:

Proposition 4.1**.**

In the setting of 3.1\wrtusdrflem:DoubleSumBound, we have

[TABLE]

where the implied constant is absolute.

Proof.

We keep the notation $\mathfrak{S}$ from the proof of 3.1\wrtusdrflem:DoubleSumBound. Then

[TABLE]

From the Weil bound (for the sum over $w$) and using Cauchy’s inequality, we infer

[TABLE]

Upon expanding the square in both sums and using orthogonality of characters, this yields

[TABLE]

which gives the result. ∎