Congruences with intervals and arbitrary sets

TL;DR

This paper establishes bounds on the number of solutions to certain congruences involving intervals and arbitrary sets in finite fields, with applications to character sums and Kloosterman sums, advancing understanding in analytic number theory.

Contribution

It provides new bounds for solutions to specific congruences in finite fields, which are optimal in many parameter ranges, and applies these bounds to improve estimates of character and Kloosterman sums.

Findings

Bound on solutions to congruences involving intervals and arbitrary sets.

Optimal bounds in a wide parameter range.

Improved estimates for character sums and Kloosterman sums.

Abstract

Given a prime $p$, an integer $H\in[1,p)$, and an arbitrary set $\cal M\subseteq \mathbb F_p^*$, where $\mathbb F_p$ is the finite field with $p$ elements, let $J(H,\cal M)$ denote the number of solutions to the congruence $$ xm\equiv yn\bmod p $$ for which $x,y\in[1,H]$ and $m,n\in\cal M$. In this paper, we bound $J(H,\cal M)$ in terms of $p$, $H$ and the cardinality of $\cal M$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski, Michel and Sawin (2018).

Equations

$$xm\equiv yn\mathrm{mod}p$$

$${\mathbb{F}}_p=\{0,\dots ,p-1\}.$$

$$xm\equiv yn\mathrm{mod}p$$

$$J(H,\mathcal{M})=\frac{H^2{M}^2}{p}+O(H{M}^{3/2}\log p)$$

$$J(H,\mathcal{M})=\frac{H^2{M}^2}{p}+O\left(HM{p}^{o(1)}\right)(p\to \infty )$$

$$J(H,\mathcal{M})⩽\left(H{p}^{-1/\ell }+1\right)H^{1+o(1)}{M}^{1+1/\ell }(H\to \infty ).$$

$$J(H,\mathcal{M})\ll \{\begin{array}{ll}{H}^2{M}^2{p}^{-1}+HM{p}^{o(1)}& ifH⩾p^{2/3},\\ H^2{M}^2{p}^{-1}+H{M}^{7/4}{p}^{-1/4+o(1)}+M^2& ifH<p^{2/3}andM⩾p^{1/3},\\ HM{p}^{o(1)}+M^2& ifH<p^{2/3}andM<p^{1/3}.\end{array}$$

$$J(H,\mathcal{M})\ll H^2{M}^2{p}^{-1}+HM{p}^{o(1)}+M^2.$$

$$H⩾p^{2/3+o(1)}\text{or}M⩽\min \{H,p^{1/3}\}{p}^{o(1)}.$$

$$W_{\chi }(H,\mathcal{K},\mathcal{M};\mathbf{\alpha },\mathbf{\zeta },\mathbf{\eta })=\sum _{h=1}^H\sum _{k\in \mathcal{K}}\sum _{m\in \mathcal{M}}{\alpha }_h{\zeta }_k{\eta }_m\chi (h+km).$$

$$|W_{\chi }(H,\mathcal{K},\mathcal{M};\mathbf{\alpha },\mathbf{\zeta },\mathbf{\eta })|$$

$$⩽HKM\left(p^{-1/2\ell }+H^{-1/2\ell }{M}^{-1/2\ell }+H^{-1/\ell }\right)\left(p^{1/4\ell }+K^{-1/2}{p}^{1/2\ell }\right)p^{o(1)}.$$

$$|W_{\chi }(H,\mathcal{K},\mathcal{M};\mathbf{\alpha },\mathbf{\zeta },\mathbf{\eta })|$$

$$⩽HKM\left(p^{-1/4\ell }+(HM{p}^{-1/2}{)}^{-1/2\ell }+(H{p}^{-1/4}{)}^{-1/\ell }\right)p^{o(1)}.$$

$$K⩾p^{\epsilon },M⩽p^{1/3+o(1)},HM⩾p^{1/2+\epsilon }\text{mbox}andH⩾p^{1/4+\epsilon }.$$

$$W_{\chi }(H,\mathcal{K},\mathcal{M};\mathbf{\alpha },\mathbf{\zeta },\mathbf{\eta })\ll HKM{p}^{-\kappa }.$$

$$S_{\chi }(H,\mathcal{M})=\sum _{m\in \mathcal{M}}\left|\sum _{h=1}^H\chi (h+m)\right|.$$

$$K_r(n)=\frac{1}{p^{(r-1)/2}}\sum _{\begin{array}{c}{x}_1,\dots ,x_r=1\\ x_1\cdots x_r\equiv n\mathrm{mod}p\end{array}}^{p-1}{\mathbf{e}}_p(x_1+\cdots +x_r),$$

$$|K_r(n)|⩽r;$$

$${\mathcal{S}}_r(\mathbf{\alpha };\mathcal{M},\mathcal{N})=\sum _{m\in \mathcal{M}}\sum _{n\in \mathcal{N}}{\alpha }_m{K}_r(mn)$$

$${\mathcal{S}}_r(\mathbf{\alpha },\mathbf{\beta };\mathcal{M},\mathcal{N})=\sum _{m\in \mathcal{M}}\sum _{n\in \mathcal{N}}{\alpha }_m{\beta }_n{K}_r(mn)$$

$$|{\mathcal{S}}_r(\mathbf{\alpha };\mathcal{M},\mathcal{N})|$$

$$+M^{-1/2\ell }{N}^{-1/\ell }{p}^{1/2\ell +1/2{\ell }^2}+N^{-3/2\ell }{p}^{1/2\ell +1/{\ell }^2})p^{o(1)}.$$

$$|{\mathcal{S}}_r(\mathbf{\alpha };\mathcal{M},\mathcal{N})|⩽MN\left(M^{-1/2\ell }{N}^{-1/\ell }{p}^{1/2\ell +1/2{\ell }^2}\right)p^{o(1)}$$

$$p^{1/\ell }⩽N⩽\frac{1}{2}{p}^{1/2+1/2\ell }$$

$$p^{1/\ell }⩽N\text{mbox}andN{M}^{+}⩽\frac{1}{2}{p}^{1+1/2\ell },$$

$$M^{+}=\underset{m\in \mathcal{M}}{\max }m.$$

$$M{N}^2⩾p^{1+1/\ell +o(1)}⩾p^{7/6+o(1)}(\ell ⩽6);$$

$$\sum _{\chi \in \mathcal{X}}\chi (z)=\{\begin{array}{ll}p-1& \text{if z=1},\\ 0& \text{otherwise}.\end{array}$$

$$\begin{array}{rl}J(H,\mathcal{M})& =\sum _{x,y=1}^H\sum _{m,n\in \mathcal{M}}\frac{1}{p-1}\sum _{\chi \in \mathcal{X}}\chi (x{y}^{-1}m{n}^{-1})\\ & =\frac{1}{p-1}\sum _{\chi \in \mathcal{X}}{\left|\sum _{x=1}^H\chi (x)\right|}^2{\left|\sum _{m\in \mathcal{M}}\chi (m)\right|}^2,\end{array}$$

$$\max_{\chi\in\mathcal{X}^{*}}\bigg{|}\sum_{a\in\mathbb{F}_{p}}\chi(f(a))\cdot\overline{\chi}(g(a))\bigg{|}\ll\left(\deg f+\deg g\right)p^{1/2}.$$

$$KJ(H,\mathcal{M})=HJ(K,\mathcal{M})+O\left((H+K)(HK{p}^{-1}+1)M^2\right).$$

Full text

[TABLE]

for which $x,y\in[1,H]$ and $m,n\in\mathcal{M}$. In this paper, we bound $J(H,\mathcal{M})$ in terms of $p$, $H$ and the cardinality of $\mathcal{M}$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski, Michel and Sawin (2018).

1. Introduction

1.1. Set up

For each prime $p$ we denote by $\mathbb{F}_{p}$ the finite field with $p$ elements, which can be identified with the set

[TABLE]

Given an integer $H\in[1,p)$ and an arbitrary set $\mathcal{M}\subseteq\mathbb{F}_{p}^{*}$, let $J(H,\mathcal{M})$ be the number of solutions $(x,y,m,n)$ to the congruence

[TABLE]

for which $x,y\in[1,H]$ and $m,n\in\mathcal{M}$. In this paper, we study the problem of bounding $J(H,\mathcal{M})$ in terms of $p$, $H$ and the cardinality $M=|\mathcal{M}|$.

In what follows, we freely alternate between the language of congruences and that of equations over finite fields. For example, in view of the isomorphism $\mathbb{Z}/p\mathbb{Z}\cong\mathbb{F}_{p}$ the congruence (1.2) is equivalent to equation $xm=yn$ over the field $\mathbb{F}_{p}$.

Throughout, the notations $U=O(V)$ and $U\ll V$ are each equivalent to the statement that the inequality $|U|\leqslant c\,V$ holds with a constant $c>0$ which may depend (where obvious) on the integer $\ell\geqslant 1$ or the real number $\varepsilon>0$.

1.2. Initial estimates

It is straightforward to show that the estimate

[TABLE]

holds. Indeed, (1.3) follows easily from (3.2) below combined with a bound on power moments of character sums; see, for example, [1, Theorem 2]. Moreover, the same method shows that (1.3) holds for congruences in which the variables $x,y$ lie in intervals of the form $[A+1,A+H]$.

The stronger conditional estimate

[TABLE]

can be established using (3.2) in conjunction with a “square-root cancellation” bound on character sums implied by the Generalised Riemann Hypothesis; see, for example, Munsch and Shparlinski [16, Equation (4)].

Furthermore, it is shown in the proof of a result of Garaev (see the bound on $J$ in the proof of [8, Lemma 2.1]) that for any fixed integer $\ell\geqslant 1$ one has the upper bound

[TABLE]

In the present paper, we obtain several results which improve (1.3) and (1.5). Moreover, in a wide range of the parameters $p,H,M$ we achieve an unconditional upper bound on $J(H,\mathcal{M})$ that has roughly the same shape as (1.4).

As an application of this bound, we give a new estimate on certain trilinear sums of multiplicative characters. We also combine it with recent results of Kowalski, Michel and Sawin [15] to derive a new bound on bilinear sums with Kloosterman sums that extends the range of the applicability of [15].

1.3. Notational conventions

Throughout the paper, the letter $p$ always denotes a prime number.

We use $|\mathcal{A}|$ to denote the cardinality of a finite set $\mathcal{A}$.

The notations $U=O(V)$, $U\ll V$ and $V\gg U$ are all used to indicate that $|U|\leqslant c|V|$ hold for some absolute constant $c>0$, and we write $U\asymp V$ if $U\ll V\ll U$. We also use $U^{o(1)}$ to denote any function $f(U)$ such that $\log|f(U)|/\log U\to 0$ as $U\to\infty$.

2. Main results

2.1. Congruences

We use ideas of Heath-Brown [10] to establish the following estimate.

Theorem 2.1.

For an integer $H\in[1,p)$ and a set $\mathcal{M}\subseteq\mathbb{F}_{p}^{*}$ of cardinality $M$, the following holds as $p\to\infty$:

[TABLE]

For $M\leqslant p^{1/3+o(1)}$, Theorem 2.1 yields the following unconditional variant of the conditional estimate (1.4):

[TABLE]

In particular, we have (1.4) if

[TABLE]

2.2. Trilinear character sums

We use $\mathcal{X}$ to denote the set of multiplicative characters of $\mathbb{F}_{p}^{*}$, and $\mathcal{X}^{*}=\mathcal{X}\setminus\{\chi_{0}\}$ is the set of nonprincipal characters; for the relevant background on multiplicative characters, we refer the reader to Iwaniec and Kowalski [11, Chapter 3].

We now give some applications to bounds of certain trilinear character sums. Specifically, for an integer $H\in[1,p)$, two sets $\mathcal{K},\mathcal{M}\subseteq\mathbb{F}_{p}^{*}$, a character $\chi\in\mathcal{X}^{*}$, and arbitrary complex weights $\bm{\alpha}=(\alpha_{h})_{h=1}^{H}$, $\bm{\zeta}=(\zeta_{k})_{k\in\mathcal{K}}$ and $\bm{\eta}=(\eta_{m})_{m\in\mathcal{M}}$, we define

[TABLE]

Similar sums with only one set in $\mathbb{F}_{p}^{*}$ have been previously studied; a variety of bounds and corresponding applications can be found in [4, 5, 13, 18]. To simplify the formulation the next theorem (and since it is the most interesting case in view of, e.g., the results of Chang [5]) we consider only the case in which one of the sets has its cardinality bounded above by $p^{1/3+o(1)}$ (thus, the bound (2.1) is at our disposal).

Theorem 2.2.

Let the notation be as above, and let $K$ and $M$ denote the cardinalities of $\mathcal{K}$ and $\mathcal{M}$, respectively. Suppose that $M\leqslant p^{1/3+o(1)}$ and that the weights $\alpha_{h},\zeta_{k},\eta_{m}$ are all bounded by one in absolute value. Then, for any fixed integer $\ell\geqslant 1$ we have

[TABLE]

In particular, if under the conditions of Theorem 2.2 we have $K\geqslant p^{\varepsilon}$ for some fixed $\varepsilon>0$, then taking $\ell$ large enough to guarantee that $p^{1/4\ell}\geqslant K^{-1/2}p^{1/2\ell}$, the bound of Theorem 2.2 becomes

[TABLE]

Thus, we obtain the following corollary.

Corollary 2.3.

For any $\varepsilon>0$ there exists $\kappa>0$ with the following property. Let the notation be as in Theorem 2.2. Suppose that

[TABLE]

Suppose further that the weights $\alpha_{h},\zeta_{k},\eta_{m}$ are all bounded by one in absolute value. Then

[TABLE]

It is clear that Corollary 2.3, coupled with standard techniques using bivariate shifts $h\mapsto h+uv$, can be used to obtain nontrivial estimates for double sums

[TABLE]

2.3. Bilinear sums of Kloosterman sum

Next, we consider multidimensional Kloosterman sums of the form

[TABLE]

where ${\mathbf{\,e}}(t)=e^{2\pi it/p}$ for all $t\in\mathbb{R}$. By the classical Deligne bound we have

[TABLE]

see, for example, [11, Equation (11.58)] and the follow-up discussion.

Recently, motivated by an abundance of applications, there has been considerable interest in the estimation of weighted sums of Kloosterman sums

[TABLE]

with two sets $\mathcal{M},\mathcal{N}\subseteq\mathbb{F}_{p}^{*}$ and complex weights $\bm{\eta}=\left(\eta_{m}\right)_{m\in\mathcal{M}}$, and also

[TABLE]

with complex weights $\bm{\alpha}=\left(\alpha_{m}\right)_{m\in\mathcal{M}}$ and $\bm{\beta}=\left(\beta_{n}\right)_{n\in\mathcal{N}}$; see [2, 3, 6, 14, 15, 17, 19, 20] and the references therein. Of course, only bounds that are superior to those that follow directly from (2.2) are of interest and use.

The bound of Theorem 2.1 can be embedded in the arguments of [2, 14, 15] which, for both sums $\mathscr{S}_{r}(\bm{\alpha};\mathcal{M},\mathcal{N})$ and $\mathscr{S}_{r}(\bm{\alpha},\bm{\beta};\mathcal{M},\mathcal{N})$, rely on a bound for $J(H,\mathcal{M})$ in the special that $\mathcal{M}=\{1,\ldots,M\}$ for some positive integer $M$. In the present paper, we demonstrate the idea only in the simple case of the sums $\mathscr{S}_{r}(\bm{\alpha};\mathcal{M},\mathcal{N})$.

Theorem 2.4.

For any set $\mathcal{M}\subseteq\mathbb{F}_{p}^{*}$ of cardinality $M$, an interval $\mathcal{N}\subseteq\mathbb{F}_{p}^{*}$ of length $N<p$, complex weights $\alpha_{m}$ bounded by one in absolute value, and a fixed even integer $\ell\geqslant 1$, we have

[TABLE]

Remark 2.5.

The results of [19] only apply to sums $\mathscr{S}_{r}(\bm{\alpha};\mathcal{M},\mathcal{N})$ with $r=2$ (i.e., to classical one dimensional Kloosterman sums); they hold arbitrary sets $\mathcal{M}\subseteq\mathbb{F}_{p}^{*}$ and intervals $\mathcal{N}\subseteq\mathbb{F}_{p}^{*}$ of length $N<p$. Moreover, in the special case that $\mathcal{M}=\{1,\ldots,M\}$, [19, Theorem 2.1] gives the strongest known bound in the crucial range where $M$ and $N$ are both of size $p^{1/2+o(1)}$. We remark, however, that if $M$ and $N$ are comparable in size, then the bound of [19, Theorem 2.1] is nontrivial only if both quantities exceed $p^{4/9+\varepsilon}$ for some fixed $\varepsilon>0$, whereas Theorem 2.4 is nontrivial once both quantities exceed $p^{1/3+\varepsilon}$. ∎

Remark 2.6.

Assuming that $|\alpha_{m}|\leqslant 1$ for all $m\in\mathcal{M}$, the bounds of [15, Theorem 4.2] imply that for any fixed integer $\ell\geqslant 1$, the bound

[TABLE]

holds provided that the integers $N$ and the set $\mathcal{M}\subseteq\mathbb{F}_{p}^{*}$ satisfy at least one of the two conditions

[TABLE]

or

[TABLE]

where (recalling the convention (1.1)) we denote

[TABLE]

In fact, the proof of [15, Theorem 4.2] can be easily extended to cover arbitrary intervals $\mathcal{N}=\{s+1,\ldots,s+N\}$ (not only initial intervals of the form $\{1,\ldots,N\}$) and arbitrary sets $\mathcal{M}\subseteq\mathbb{F}_{p}^{*}$. Indeed, in the case (2.4) it is enough to use a result of Ayyad, Cochrane and Zheng [1, Theorem 1] in the appropriate place (where the congruence $a_{1}n_{2}\equiv a_{2}n_{1}\bmod p$ is replaced with an equation $a_{1}n_{2}=a_{2}n_{1}$ over $\mathbb{Z}$), whereas in the case (2.5) no changes are required.

Furthermore, under (2.4) the argument in the proof of [15, Theorem 4.2] applies to arbitrary sets $\mathcal{M}$, but in the case of the condition (2.5) the restriction on the size of $M^{+}$ is crucial.

Our Theorem 2.4 gives a somewhat weaker bound than (2.3), but it applies in greater generality without any restriction on $M^{+}$.

To illustrate this, let $\mathcal{M}\subseteq\mathbb{F}_{p}^{*}$ be such that $M^{+}>p^{1/2}$; then both (2.4) and (2.5) require that $N\leqslant\tfrac{1}{2}p^{1/2+1/2\ell}$. If $N=p^{15/26+o(1)}$ (say), then the bound (2.3) only applies if $\ell\leqslant 6$, so it yields a nontrivial bound only if

[TABLE]

this requires that $M\geqslant p^{1/78+o(1)}$. On the other hand, Theorem 2.4, for an appropriate choice of $\ell$ (which is not restricted by any conditions similar to (2.4) and (2.5)) provides a nontrivial bound already for $M\geqslant p^{\varepsilon}$. ∎

Remark 2.7.

As in [14, 15], we expect that Theorem 2.4 can be extended to a broad class of trace functions satisfying suitable “big monodromy” assumptions and other natural hypotheses. ∎

3. Preparations

3.1. Character sums

For any $z\in\mathbb{F}_{p}$ we have the well known orthogonality relation (see [11, Section 3.1]):

[TABLE]

Using (3.1) we derive that

[TABLE]

which is used in our proof of Theorem 2.1.

We also need the Weil bound on multiplicative character sums in the following form; see [11, Theorem 11.23].

Lemma 3.1.

For square-free and coprime polynomials $f,g\in\mathbb{F}_{p}[X]$ we have

[TABLE]

3.2. Recursive inequality

In the proof of Theorem 2.1, the following lemma is used to bound $J(H,\mathcal{M})$ in terms of $J(K,\mathcal{M})$ with a suitably chosen $K$.

Lemma 3.2.

For any integers $H,K\in[1,p)$ we have the uniform estimate

[TABLE]

Proof.

By symmetry, we can assume that $K\leqslant H$.

For a fixed pair $(m,n)\in\mathcal{M}^{2}$, let $\Lambda(m,n)$ be the lattice defined by

[TABLE]

which clearly has determinant $\det\Lambda(m,n)=p$. By [9, Lemma 1] there are basis vectors ${\mathbf{e}},{\mathbf{f}}\in\Lambda(m,n)$ such that the Euclidean lengths $\|{\mathbf{e}}\|$ and $\|{\mathbf{f}}\|$ satisfy

[TABLE]

and such that for any vector ${\mathbf{u}}=a\,{\mathbf{e}}+b\,{\mathbf{f}}$ with $a,b\in\mathbb{Z}$ one has

[TABLE]

with some absolute constant $c_{0}>0$.

First, we consider pairs $(m,n)$ in $\mathcal{M}^{2}$ for which there exist ${\mathbf{e}},{\mathbf{f}}\in\Lambda(m,n)$ as above with $\|{\mathbf{f}}\|\leqslant 2^{1/2}c_{0}H$. Using (3.4) and (3.5), the total number of vectors ${\mathbf{u}}=(x,y)\in\Lambda(m,n)$ with $x,y\in[1,H]$ is at most

[TABLE]

It follows that the contribution to $J(H,\mathcal{M})$ from all such pairs is $O(H^{2}M^{2}p^{-1})$. Similarly, the number of vectors ${\mathbf{u}}=(x,y)\in\Lambda(m,n)$ with $x,y\in[1,K]$ is at most

[TABLE]

since

[TABLE]

Hence, the contribution to $J(K,\mathcal{M})$ from all such pairs is $O(HKM^{2}p^{-1}+M^{2})$.

Let $\mathscr{M}$ be the set formed from the remaining pairs $(m,n)$ in $\mathcal{M}^{2}$. For each pair in $\mathscr{M}$, fix a choice of basis vectors ${\mathbf{e}},{\mathbf{f}}\in\Lambda(m,n)$ satisfying (3.4) and (3.5). Taking into account that $\|{\mathbf{f}}\|>2^{1/2}c_{0}H$, if

[TABLE]

for some $x,y\in[1,H]$ and $a,b\in\mathbb{Z}$, then $b=0$ by (3.5); in other words,

[TABLE]

Writing ${\mathbf{\,e}}=(r,s)$, this implies that the numbers $r,s$ have the same sign, and $rs\neq 0$. Replacing ${\mathbf{e}}$ by $-{\mathbf{e}}$ we can assume $r,s\geqslant 1$. Now one sees easily that the number of vectors ${\mathbf{u}}$ satisfying (3.6) is precisely $\left\lfloor H/\|{\mathbf{e}}\|_{\infty}\right\rfloor$, where $\|{\mathbf{e}}\|_{\infty}=\max\{r,s\}$.

We now use ${\mathbf{e}}_{m,n}$ to denote the corresponding vector ${\mathbf{e}}$ coming from a given pair $(m,n)\in\mathscr{M}$. Thus, the contribution to $J(H,\mathcal{M})$ from the pairs in $\mathscr{M}$ is

[TABLE]

Since $H\geqslant K$, a similar result holds with $H$ replaced by $K$.

Combining all of the above results, we deduce the following estimates:

[TABLE]

and these imply (3.3) for $K\leqslant H$. ∎

4. Proofs of Main Results

4.1. Proof of Theorem 2.1

We first bound $J(K,\mathcal{M})$ with a suitably chosen $K$, and then we apply Lemma 3.2.

Replacing $H$ with $K$ in (3.2) and using the Cauchy inequality, we derive that

[TABLE]

where we have used (3.1) in the second step. Since $\chi$ is multiplicative this implies

[TABLE]

where $L=K^{2}$, and $\rho(u)$ is the number of pairs $(x,y)\in[1,K]^{2}$ such that $u=xy$. Taking into account (3.1) once again, we see that

[TABLE]

where

[TABLE]

Since $L\leqslant p^{2}$, using a trivial bound on the divisor function (see, for example, [11, Equation (1.81)]) we have

[TABLE]

Combining this bound with (4.1) and (4.2), it follows that

[TABLE]

and so by Lemma 3.2 it follows that

[TABLE]

In the case that $H\geqslant p^{2/3}$, we take $K=\left\lfloor H^{1/2}\right\rfloor$. Since $L\leqslant H$, we conclude from (4.3) that

[TABLE]

which implies that

[TABLE]

This proves the theorem in this case.

Next, suppose that $H<p^{2/3}$ and $M\geqslant p^{1/3}$. Let $K=\left\lceil M^{1/4}p^{1/4}\right\rceil$. Since $K\geqslant p^{1/3}$ and so $L\geqslant p^{2/3}$, using (4.4) with $H=L$ we see that

[TABLE]

hence by (4.3) we have

[TABLE]

The third term is dominated by the second term since $M<p$, thus it can be dropped, and the theorem is proved in this case.

Finally, suppose that $H<p^{2/3}$ and $M<p^{1/3}$. Put $K=\left\lceil p^{1/3}\right\rceil$. Since $L\geqslant p^{2/3}$, using (4.4) with $H=L$ it follows that

[TABLE]

hence by (4.3), our choice of $K$, and the fact that $HM<p$, the theorem follows in this case. This concludes the proof.

4.2. Proof of Theorem 2.2

For each $\lambda\in\mathbb{F}_{p}^{*}$ let

[TABLE]

Then, using the multiplicativity of $\chi$, we have

[TABLE]

Clearly,

[TABLE]

For any fixed integer $\ell\geqslant 1$ we write

[TABLE]

and applying the Hölder inequality, obtaining

[TABLE]

where $\overline{\chi}$ is the conjugate character.

We now apply Lemma 3.1 to the sums over $\lambda$ when the sets

[TABLE]

are different, and we use the trivial bound otherwise; this leads to the bound

[TABLE]

Since $M\leqslant p^{1/3+o(1)}$, we can apply (2.1) to derive that

[TABLE]

and the result follows.

4.3. Proof of Theorem 2.4

In what follows, we use notation of the form $x\sim X$ as an abbreviation for $X<x\leqslant 2X$.

To prove Theorem 2.4, we follow the strategy of the proof of [15, Theorem 4.2], which is summarised in [15, §4.2]. Let $A$ and $B$ be integer parameters for which

[TABLE]

Then, as in [15, Section 4.2] and [7, Chapter IV] we have

[TABLE]

with some complex weights $\bm{\eta}=(\eta_{b})_{b\sim B}$ of absolute value one, and

[TABLE]

where $\mathcal{A}$ denotes the set of integers $a\sim A$.

Following [15, Section 4.2] (see also the proof of Theorem 2.2), we use the Hölder inequality to derive that

[TABLE]

for any fixed integer $\ell\geqslant 1$, where

[TABLE]

and

[TABLE]

As in [15] we have trivially

[TABLE]

and also

[TABLE]

There are at most $J(2A,\mathcal{M})$ solutions $(a_{1},a_{2},m_{1},m_{2})\in\mathcal{A}^{2}\times\mathcal{M}^{2}$ to the congruence $a_{1}m_{1}\equiv a_{2}m_{2}\bmod p$, and for any such solution, there are at most $N$ pairs $(n_{1},n_{2})\in\mathcal{N}^{2}$ such that $a_{1}n_{1}\equiv a_{2}n_{2}\bmod p$; therefore,

[TABLE]

Substituting (4.7) and (4.8) in (4.6), we obtain that

[TABLE]

We turn our attention to the sum $S_{K}$. Estimating $S_{K}$ lies at the heart of the method of [15, Section 4.2], where the bound

[TABLE]

with some fixed $\gamma\geqslant 0$ is derived from [15, Lemma 2.3] and [15, Theorem 4.4]. Moreover, from the statement and proof of [15, Theorem 4.3] we see that the way $\gamma$ is defined, the product $\gamma\ell$ is an integer not less than $(\ell-1)/2$; for even $\ell$ this implies that $\gamma\geqslant 1/2$, and so

[TABLE]

Thus, (4.10) simplifies to

[TABLE]

Taking $B=\left\lfloor p^{1/\ell}\right\rfloor$, the bound (4.11) becomes $S_{K}\ll B^{2\ell}p$; using this bound in (4.9) along with the bound for $J(2A,\mathcal{M})$ afforded by Theorem 2.1, we get that

[TABLE]

We now choose

[TABLE]

which guarantees that the condition (4.5) is met, and after simple calculations we obtain the stated bound.

5. Comments

Iwaniec and Sárközy [12] have considered a question about the distance between the product set of two sufficiently dense sets of integers in the interval $[1,N]$ and the set of perfect squares. The same question can also be considered modulo $p$, which immediately leads to the question of obtaining nontrivial bounds on the trilinear sums $W_{\chi}(H,\mathcal{K},\mathcal{M};\bm{\alpha},\bm{\zeta},\bm{\eta})$ as in Section 2.2 with a quadratic character $\chi$.

Acknowledgements

The authors are grateful to Roger Heath-Brown for several very useful discussions and for making available a preliminary version of [10].

This work was supported in part by the Australian Research Council Grant DP170100786 (for I. E. Shparlinski).