Multiplicative Energy of Shifted Subgroups and Bounds On Exponential Sums with Trinomials in Finite Fields

TL;DR

This paper introduces new bounds on colinear triples in subgroups of prime finite fields and applies these results to improve bounds on exponential sums involving trinomials.

Contribution

It provides novel bounds on colinear triples in subgroups and leverages these to enhance estimates on exponential sums with trinomials in finite fields.

Findings

New bounds on colinear triples in subgroups of prime finite fields

Improved bounds on exponential sums with trinomials

Enhanced understanding of additive and multiplicative structures in finite fields

Abstract

We give a new bound on colinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.

Equations

$$\Psi (X)=\sum _{i=1}^t{a}_i{X}^{k_i}$$

$$S_{\chi }(\Psi )=\sum _{x\in {\mathbb{F}}_p^{\ast }}\chi (x){\mathbf{e}}_p(\Psi (x)),$$

$$|S_{\chi }(\Psi )|\le \max \{k_1,\dots ,k_t\}{p}^{1/2},$$

$$\Psi (X)=a{X}^k+b{X}^{\ell }+c{X}^m,$$

$$\Psi (X)=a{X}^k+F(X^{\ell })+G(X^m)$$

$$S_{\chi }(\Psi )\ll {\left(\frac{k\ell m}{\max \{k,\ell ,m\}}\right)}^{1/4}{p}^{7/8}$$

$$S_{\chi }(\Psi )\ll (k\ell m{)}^{1/9}{p}^{5/6}$$

$$S_{\chi }(\Psi )\ll D^{1/2}{p}^{7/8}+(k\ell m{)}^{1/4}{p}^{5/8}$$

$$\frac{u_1-{\lambda }_1{v}_1}{u_1-{\lambda }_1{w}_1}=\frac{u_2-{\lambda }_2{v}_2}{u_2-{\lambda }_2{w}_2},u_i,v_i,w_i\in {\mathcal{U}}_i,i=1,2.$$

$$\mathrm{T}(\mathcal{U})={\mathrm{T}}_{1,1}(\mathcal{U},\mathcal{U}).$$

$$\mathrm{T}(\mathcal{U})=\frac{|\mathcal{U}{|}^6}{p}+O\left(p^{1/2}|\mathcal{U}{|}^{7/2}\right)$$

$${\mathrm{T}}_{{\lambda }_1,{\lambda }_2}({\mathcal{U}}_1,{\mathcal{U}}_2)=\frac{|{\mathcal{U}}_1{|}^3|{\mathcal{U}}_2{|}^3}{p}+O\left(p^{1/2}|{\mathcal{U}}_1{|}^{3/2}|{\mathcal{U}}_2{|}^2+|{\mathcal{U}}_1{|}^3|{\mathcal{U}}_2|\right),$$

$${\mathrm{T}}_{\lambda}({\mathcal{G}})-\frac{|{\mathcal{G}}|^{6}}{p}\ll\left\{\begin{array}[]{ll}p^{1/2}|{\mathcal{G}}|^{7/2},&\text{if $|{\mathcal{G}}|\geq p^{2/3}$},\\ |{\mathcal{G}}|^{5}p^{-1/2},&\text{if $p^{2/3}>|{\mathcal{G}}|\geq p^{1/2}\log p$},\\ |{\mathcal{G}}|^{4}\log|{\mathcal{H}}|,&\text{if $|{\mathcal{G}}|<p^{1/2}\log p$}.\end{array}\right.$$

$$d=\gcd (k,p-1),e=\gcd (\ell ,p-1),f=\gcd (m,p-1)$$

$$g=\frac{d}{\gcd (d,f)},h=\frac{e}{\gcd (e,f)}.$$

$$S_{\chi}(\Psi)\ll\left\{\begin{array}[]{ll}p^{7/8}f^{1/8},&\text{if $h\geq\left(p\log p\right)^{1/2}$},\\ p^{15/16}(f/h)^{1/8}\left(\log p\right)^{1/16},&\text{if $g\geq\left(p\log p\right)^{1/2}>h$},\\ p(f/gh)^{1/8}\left(\log p\right)^{1/8},&\text{if $g<\left(p\log p\right)^{1/2}$}.\end{array}\right.$$

$$p^{\delta }<f<(de{)}^{1-\delta }$$

$$S_{\chi }(\Psi )\ll p(f/gh{)}^{1/8+o(1)}=p(f/de{)}^{1/8+o(1)},$$

$$|\Theta |\le \min \left\{|\mathcal{G}||\mathcal{H}|,\frac{p^3}{|\mathcal{G}{|}^2|\mathcal{H}{|}^2}\right\}$$

$$\sum _{(u,v)\in \Theta }\left|\{(x,y)\in \mathcal{G}\times \mathcal{H}:ux+vy=1\}\right|\ll (|\mathcal{G}||\mathcal{H}||\Theta {|}^2{)}^{1/3}.$$

$$\sum _{(u,v)\in \Theta }\left|\{(x,y)\in \mathcal{G}\times \mathcal{H}:ux+vy=1\}\right|\ll \min \{|\mathcal{G}|,|\mathcal{H}|\}|\Theta |.$$

$${\ell }_{a,b}=\{(x,y)\in {\mathbb{F}}_p^2:y=ax+b\}$$

$${\iota }_{\mathcal{A},\mathcal{B}}\left({\ell }_{a,b}\right)=\left|{\ell }_{a,b}\cap \left(\mathcal{A}\times \mathcal{B}\right)\right|.$$

$$\sum _{(a,b)\in {\mathbb{F}}_p^2}{\iota }_{\mathcal{A},\mathcal{B}}\left({\ell }_{a,b}\right)=\sum _{(a,b)\in {\mathbb{F}}_p^2}{\iota }_{\mathcal{A},\mathcal{B}}\left({\ell }_{\lambda a,\mu b}\right)=p|\mathcal{A}||\mathcal{B}|$$

$$\sum _{(a,b)\in {\mathbb{F}}_p^2}{\iota }_{\mathcal{A},\mathcal{B}}\left({\ell }_{a,b}\right){\iota }_{\mathcal{A},\mathcal{B}}\left({\ell }_{\lambda a,\mu b}\right)=|\mathcal{A}{|}^2|\mathcal{B}{|}^2-|\mathcal{A}||\mathcal{B}{|}^2+p|\mathcal{A}||\mathcal{B}|.$$

$$\sum _{(a,b)\in {\mathbb{F}}_p^2}$$

$$=\sum _{(u,v,x,y)\in \mathcal{A}\times \mathcal{B}\times \mathcal{A}\times \mathcal{B}}\left|\{(a,b)\in {\mathbb{F}}_p^2:v=au+b,y=\lambda ax+\mu b\}\right|.$$

$$(u,v)=(\lambda {\mu }^{-1}x,{\mu }^{-1}y)$$

$$|\mathcal{A}{|}^2|\mathcal{B}{|}^2-|\mathcal{A}||\mathcal{B}|\left(|\mathcal{B}|-1\right)-|\mathcal{A}||\mathcal{B}|=|\mathcal{A}{|}^2|\mathcal{B}{|}^2-|\mathcal{A}||\mathcal{B}{|}^2$$

$$\sum _{(a,b)\in {\mathbb{F}}_p^2}{\left({\iota }_{\mathcal{A},\mathcal{B}}\left({\ell }_{a,b}\right)-\frac{|\mathcal{A}||\mathcal{B}|}{p}\right)}^2\le p|\mathcal{A}||\mathcal{B}|.$$

$${\mathrm{T}}_{\lambda ,\mu }(\mathcal{A},\mathcal{B})=\sum _{(a,b)\in {\mathbb{F}}_p^2}{\iota }_{\mathcal{A},\mathcal{B}}\left({\ell }_{a,b}\right){\iota }_{\mathcal{A},\mathcal{B}}{\left({\ell }_{\lambda a,\mu b}\right)}^2+O(|\mathcal{A}{|}^2|\mathcal{B}{|}^2).$$

Full text

1. Introduction

1.1. Set up

For a prime $p$, we use ${\mathbb{F}}_{p}$ to denote the finite field of $p$ elements.

For a $t$-sparse polynomial

[TABLE]

with some pairwise distinct positive integer exponents $k_{1},\ldots,k_{t}$ and coefficients $a_{1},\ldots,a_{t}\in{\mathbb{F}}_{p}^{*}$, and a multiplicative character $\chi$ of ${\mathbb{F}}_{p}^{*}$ we define the sums

[TABLE]

where ${\mathbf{\,e}}_{p}(u)=\exp(2\pi iu/p)$ and $\chi$ is an arbitrary multiplicative character of ${\mathbb{F}}_{p}^{*}$. Certainly, the most interesting and well-studied special case is when $\chi=\chi_{0}$ is a principal character. However most of our results extend to the general case without any loss of strength or complication of the argument, so this is how we present them.

The main challenge here is to estimate these sums better than by the Weil bound

[TABLE]

see [36, Appendix 5, Example 12], by taking advantage of sparsity and also of the arithmetic structure of the exponents $k_{1},\ldots,k_{t}$.

For monomials $\Psi(X)=aX^{k}$ (where we can always assume that $k\mid p-1$) the first bound of this type is due to Shparlinski [32], which has then been improved and extended in various directions by Bourgain, Glibichuk and Konyagin [6], Bourgain [3], Heath-Brown and Konyagin [19], Konyagin [21], Shkredov [27], Shteinikov [34].

Akulinichev [1] gives several bounds on binomials, see also [38]. Cochrane, Coffelt and Pinner, see [8, 9, 10, 11, 12, 13] and references therein, have given a series of other bounds on exponential sums with sparse polynomials, some of which we present below in Section 1.2.

We also remark that exponential sums with sparse polynomials and a composite denominator have been studied in [4, 33].

Here we use a slightly different approach to improve some of the previous results. Our approach is reliant on reducing bounds of exponential sums with sparse polynomials to bounds of weighted multilinear exponential sums of the type considered in [25]. However, instead of applying the results of [25] directly, we first obtain a more precise variant for triple weighted sums over multiplicative subgroups of ${\mathbb{F}}_{p}^{*}$, which could be of independent interest, see Lemma 3.5 below.

This result rests on an extension of the bound on the number of collinear triples in multiplicative subgroups from [28, Proposition 1] to subgroups of any size, see Theorem 1.2. In turn, this gives a new bound on the multiplicative energy of arbitrary subgroups, see Corollary 4.1, and has several other applications, see Section 4.

Although here we concentrate on the case of trinomials

[TABLE]

our method works, without any changes, for more general sums with polynomials of the shape

[TABLE]

with arbitrary polynomials $F,G\in{\mathbb{F}}_{p}[X]$ (uniformly in the degrees of $F$ and $G$, which essentially means that they can be any functions defined on ${\mathbb{F}}_{p}$).

One can certainly use our approach for sums with quadrinomials reducing it to quadrilinear sums and using our Lemma 3.3 in an appropriate place of the argument of the proof of [25, Theorem 1.4]. Furthermore, using results of [4, 5, 17], one can consider the case of arbitrary sparse polynomials.

The notation $A\ll B$ is equivalent to $|A|\leq c|B|$ for some constant $c$, which, throughout the paper may only depend on the number of monomials in the sparse polynomials under considerations.

1.2. Previous results

We compare our results for trinomials (1.1) with the estimates of Cochrane, Coffelt and Pinner [8, Equation (1.6)]

[TABLE]

which is non-trivial for $\min\{k\ell,km,\ell m\}<p^{1/2}$, and of Cochrane and Pinner [10, Theorem 1.1]:

[TABLE]

which is non-trivial for $k\ell m<p^{3/2}$.

We also recall the bound of Cochrane, Coffelt and Pinner [9, Corollary 1.1]

[TABLE]

where $D=\gcd(k,\ell,m,p-1)$, which is non-trivial for $k\ell m<p^{3/2}$ and $D<p^{1/4}$.

1.3. New results

The following quantity is one of our main objects of study.

Definition 1.1** (Collinear triples).**

For sets ${\mathcal{U}}_{1},{\mathcal{U}}_{2}\subseteq{\mathbb{F}}^{*}_{p}$ and elements $\lambda_{1},\lambda_{2}\in{\mathbb{F}}^{*}_{p}$ we define ${\mathrm{T}}_{\lambda_{1},\lambda_{2}}({\mathcal{U}}_{1},{\mathcal{U}}_{2})$ to be the number of solutions to

[TABLE]

We also set

[TABLE]

As the relation (2.1) shows, the triples $(u_{i},v_{i},w_{i})$, $i=1,2$ satisfying (1.5) define their collinear points. Recent results on the quantity ${\mathrm{T}}({\mathcal{U}})$ for an arbitrary set ${\mathcal{U}}$ can be found in [24], where, in particular, the bound

[TABLE]

is given. This bound has been generalised in [22] as

[TABLE]

provided that $|{\mathcal{U}}_{1}|\geq|{\mathcal{U}}_{2}|$.

Note that in (1.5), as well as in all similar expressions of this type, we consider only the values of the variables for which these expressions are defined (that is, $u_{i}\neq\lambda_{i}w_{i}$, $i=1,2$ in (1.5)).

We begin by providing a new result on the number of collinear triples in subgroups. More generally, for a multiplicative subgroup ${\mathcal{G}}$ of ${\mathbb{F}}^{*}_{p}$ we define ${\mathrm{T}}_{\lambda}({\mathcal{G}})={\mathrm{T}}_{1,\lambda}({\mathcal{G}})$ which is our main object of study.

Theorem 1.2**.**

Let ${\mathcal{G}}$ be a multiplicative subgroup of ${\mathbb{F}}^{*}_{p}$. Then for any $\lambda\in{\mathbb{F}}^{*}_{p}$, we have

[TABLE]

Remark 1.3**.**

Theorem 1.2 is new only for subgroups of intermediate size $p^{2/3}>|{\mathcal{G}}|>p^{1/2}$, otherwise it is contained in [28, Proposition 1], see also Lemma 2.6 below, or in the bound (1.6) from [22].

Remark 1.4**.**

The method of proof of Theorem 1.2 also works without any changes for ${\mathrm{T}}_{\lambda,\mu}({\mathcal{G}},{\mathcal{H}})$ with two multiplicative subgroups, similarly to Lemma 2.6. However, for subgroups of significantly different sizes the optimisation part becomes rather tedious.

We use Theorem 1.2 to obtain the following new bound on trinomial sums.

Theorem 1.5**.**

Let $\Psi(X)$ be a trinomial of the form (1.1) with $a,b,c\in{\mathbb{F}}_{p}^{*}$. Define

[TABLE]

and

[TABLE]

Suppose $f\geq g\geq h$, then

[TABLE]

Note that the assumption $f\geq g\geq h$ of Theorem 1.5 does not present any additional restriction on the class of polynomials to which it applies as the roles of $k$, $\ell$ and $m$ are fully symmetric: if $h>g$, say, one can simply interchange $g$ and $h$ in the bound.

We observe that the bound of Theorem 1.5 does not directly depend on the size of the exponents $k$, $\ell$ and $m$ but rather on various greatest common divisors. In particular, it is strongest for large $d$ and $e$ and small greatest common divisors $f$, $\gcd(d,f)$ and $\gcd(e,f)$. Furthermore, it may remain nontrivial even for polynomials of very large degrees, while the bounds (1.2), (1.3) and (1.4) all become trivial for trinomials of large degree. Thus it is easy to give various families of parameters where Theorem 1.5 improves the bounds (1.2), (1.3) and (1.4) simultaneously. For example, we assume that $f>d>e$ are relatively prime positive integers with, say,

[TABLE]

for some fixed real $\delta>0$. Then $g=d$ and $h=e$ and we also have $d<p^{1/2}$, $e<p^{1/3}$. Hence, the bound of Theorem 1.5 becomes

[TABLE]

which always gives a power saving against the trivial bound. On the other hand, choosing $k$, $\ell$ and $m$ as large multiples of $d$, $e$ and $f$, respectively, say, with $k,m,\ell\geq p^{1/2+\delta}$, we see that all bounds from Section 1.2, and of course the Weil bound, are trivial.

We also give further applications of Theorem 1.2 to some additive problems with multiplicative subgroups of ${\mathbb{F}}_{p}^{*}$ in Section 4. In particular, in Corollary 4.4 we consider a modular version of the Romanoff theorem and show that for almost all primes $p$, any residue class modulo $p$ can be represented as a sum of a prime $\ell<p$ and three powers of any fixed integer $g\geq 2$. We recall that the classical result of Romanoff [26] asserts that for any fixed integer $g\geq 2$ a positive proportion of integers can be written in the form $\ell+g^{k}$, with some prime $\ell$ and non-negative integer $k$. By a result of Crocker [14], there are infinitely many positive integers not of the form $\ell+2^{k}+2^{m}$. The case of three powers of $2$ or any other base $g>2$ is widely open.

2. Collinear Triples

2.1. Prelimaries

We require some previous results. We note that we use Lemma 2.1 only for ${\mathcal{G}}={\mathcal{H}}$, however we present it and also some other results in full generality as we believe they may find several other applications and this deserves to be known better.

The first one is a result of Mit’kin [23, Theorem 2] extending that of Heath-Brown and Konyagin [19, Lemma 5], see also [21, 31] for further generalisations.

Lemma 2.1**.**

Let ${\mathcal{G}}$ and ${\mathcal{H}}$ be subgroups of ${\mathbb{F}}_{p}^{*}$ and let ${\mathcal{M}}_{\mathcal{G}}$ and ${\mathcal{M}}_{\mathcal{H}}$ be two complete sets of distinct coset representatives of ${\mathcal{G}}$ and ${\mathcal{H}}$ in ${\mathbb{F}}_{p}^{*}$. For an arbitrary set $\varTheta\subseteq{\mathcal{M}}_{\mathcal{G}}\times{\mathcal{M}}_{\mathcal{H}}$ such that

[TABLE]

we have

[TABLE]

Note that there is a natural bijection between ${\mathcal{M}}_{\mathcal{G}}$, ${\mathcal{M}}_{\mathcal{H}}$ and some subsets of the factor groups ${\mathbb{F}}_{p}^{*}/{\mathcal{G}}$ and ${\mathbb{F}}_{p}^{*}/{\mathcal{H}}$. So, one can think of $\Theta$ as a subset of ${\mathbb{F}}_{p}^{*}/{\mathcal{G}}\times{\mathbb{F}}_{p}^{*}/{\mathcal{H}}$.

Clearly, the trivial bound on the sum of Lemma 2.1 is

[TABLE]

Hence if, for example, ${\mathcal{G}}={\mathcal{H}}$, then Lemma 2.1 always significantly improves this bound.

Given a line

[TABLE]

for some pair $(a,b)\in{\mathbb{F}}_{p}^{2}$ and sets ${\mathcal{A}},{\mathcal{B}}\subseteq{\mathbb{F}}_{p}$, we denote

[TABLE]

The following elementary identities are well-known and no doubt have appeared, implicitly and explicitly, in a number of works.

Lemma 2.2**.**

Let ${\mathcal{A}},{\mathcal{B}}\subseteq{\mathbb{F}}_{p}$ and $\lambda,\mu\in{\mathbb{F}}_{p}^{*}$. Then

[TABLE]

and

[TABLE]

Proof.

The first relation is obvious as for every $(x,y,a)\in{\mathcal{A}}\times{\mathcal{B}}\times{\mathbb{F}}_{p}$ there is a unique $b=y-ax$ counted in that sum.

For the second sum, we write

[TABLE]

We now note that the $|{\mathcal{A}}||{\mathcal{B}}|$ quadruples $(u,v,x,y)\in{\mathcal{A}}\times{\mathcal{B}}\times{\mathcal{A}}\times{\mathcal{B}}$ with

[TABLE]

define exactly $p$ pairs $(a,b)=(a,v-au)\in{\mathbb{F}}_{p}^{2}$ as above. Furthermore, the $|{\mathcal{A}}||{\mathcal{B}}|\left(|{\mathcal{B}}|-1\right)$ quadruples $(u,v,x,y)\in{\mathcal{A}}\times{\mathcal{B}}\times{\mathcal{A}}\times{\mathcal{B}}$ with $u=\lambda\mu^{-1}x$ but $v\neq\mu^{-1}y$ do not define any pairs $(a,b)$ as above. The remaining

[TABLE]

pairs (including the one with $u\neq\lambda\mu^{-1}x$ but $v=\mu^{-1}y$) define one pair $(a,b)\in{\mathbb{F}}_{p}^{2}$ as above each, which concludes the proof.     \sqcap$$\sqcup

Using Lemma 2.2 with $\lambda=\mu=1$, we now immediately derive the following result:

Corollary 2.3**.**

Let ${\mathcal{A}}\subseteq{\mathbb{F}}_{p}$. Then

[TABLE]

We now link the number of collinear triples ${\mathrm{T}}_{\lambda,\mu}({\mathcal{A}},{\mathcal{B}})$ with the quantities $\iota_{{\mathcal{A}},{\mathcal{B}}}\left(\ell_{a,b}\right)$.

Lemma 2.4**.**

Let ${\mathcal{A}},{\mathcal{B}}\subseteq{\mathbb{F}}_{p}$ and $\lambda,\mu\in{\mathbb{F}}_{p}^{*}$. Then

[TABLE]

Proof.

Transforming the equation (1.5) into

[TABLE]

we introduce an error of magnitude $O(|{\mathcal{A}}|^{2}|{\mathcal{B}}|^{2})$ (coming from different pairs of variables which must be distinct). Then collecting, for every $a\in{\mathbb{F}}_{p}$, the solutions with

[TABLE]

we derive:

[TABLE]

We now denote this common value by $b$ and observe that for any $(a,b)\in{\mathbb{F}}_{p}^{2}$ there are $\iota_{{\mathcal{A}},{\mathcal{B}}}\left(\ell_{a,b}\right)\iota_{{\mathcal{A}},{\mathcal{B}}}\left(\ell_{\lambda a,\mu b}\right)^{2}$ solutions to

[TABLE]

Summing over all pairs $(a,b)\in{\mathbb{F}}_{p}^{2}$, we obtain the result.     \sqcap$$\sqcup

Corollary 2.5**.**

Let ${\mathcal{A}},{\mathcal{B}}\subseteq{\mathbb{F}}_{p}$ and $\lambda,\mu\in{\mathbb{F}}_{p}^{*}$. Then

[TABLE]

Proof.

Using the identity $X^{2}=(X-Y)^{2}+2XY-Y^{2}$ with $X=\iota_{{\mathcal{A}},{\mathcal{B}}}\left(\ell_{\lambda a,\mu b}\right)$ and $Y=|{\mathcal{A}}||{\mathcal{B}}|/p$ we see that

[TABLE]

where

[TABLE]

By Lemma 2.2, after simple calculations, we have

[TABLE]

Combining this with (2.2) yields

[TABLE]

Hence, using Lemma 2.4, we obtain the result.     \sqcap$$\sqcup

Given two sets ${\mathcal{U}},{\mathcal{V}}\subseteq{\mathbb{F}}_{p}$, we define ${\mathrm{E}}^{\times}({\mathcal{U}},{\mathcal{V}})$ to be the multiplicative energy of ${\mathcal{U}}$ and ${\mathcal{V}}$, that is, the number of solutions to

[TABLE]

For ${\mathcal{U}}={\mathcal{V}}$ we also write

[TABLE]

It is easy to see that for any subgroup of ${\mathcal{G}},{\mathcal{H}}\subseteq{\mathbb{F}}_{p}^{*}$ and $\lambda,\mu\in{\mathbb{F}}_{p}^{*}$ we have

[TABLE]

where the error term $O(|{\mathcal{G}}|^{3}|{\mathcal{H}}|)$ (which is obviously negative) accounts for zero values of the linear forms in the definition of ${\mathrm{T}}_{\lambda,\mu}({\mathcal{G}},{\mathcal{H}})$.

Finally, we need the following bound for small subgroups, which is a slightly simplified form of [28, Proposition 1] combined with (2.3).

Lemma 2.6**.**

Let ${\mathcal{G}}$ be a subgroup of ${\mathbb{F}}_{p}^{*}$ with $|{\mathcal{G}}|\geq|{\mathcal{H}}|$ and $|{\mathcal{G}}||{\mathcal{H}}|<p$. Then

[TABLE]

2.2. Initial reductions

The argument below follows [28, 29].

First of all, note that Lemma 2.6 implies the required result provided $|{\mathcal{G}}||{\mathcal{H}}|<p$ while the bound (1.6) implies it for $|{\mathcal{G}}|\geq p^{2/3}$.

So it remains to consider the case

[TABLE]

Let $\Delta\geq 3$ be a parameter to be chosen later. Using Corollaries 2.3 and 2.5, we obtain

[TABLE]

where

[TABLE]

Clearly, the contribution to $W$ from lines with $ab=0$, is at most $|{\mathcal{G}}|^{4}$ as in this case $\iota_{\mathcal{G}}\left(\ell_{a,b}\right)=0$ unless $a\in{\mathcal{G}}$ or $b\in{\mathcal{G}}$, in which case $\iota_{\mathcal{G}}\left(\ell_{a,b}\right)=|{\mathcal{G}}|$. Therefore,

[TABLE]

Thus

[TABLE]

where

[TABLE]

which is the sum we now consider.

Returning to (1.5), we see that the quantity ${\mathrm{T}}_{\lambda}({\mathcal{G}})$ is equal, up to the error $O(|{\mathcal{G}}|^{4})$, which can be absorbed in the same error term that is already present in (2.4), to the number of solutions of the equation

[TABLE]

2.3. Sets $\varTheta_{\tau}$ and ${\mathcal{Q}}_{\tau}$

Let, as before, ${\mathcal{M}}_{\mathcal{G}}$ be a set of distinct coset representatives of ${\mathcal{G}}$ in ${\mathbb{F}}_{p}^{*}$. Take another parameter $\tau\geq\Delta$ and put

[TABLE]

In other words, $\varTheta_{\tau}$ is the set of $(\alpha,\beta)\in{\mathcal{M}}_{\mathcal{G}}^{2}$ for which the lines

[TABLE]

have the intersection with ${\mathcal{G}}^{2}$ of size at least

[TABLE]

In particular,

[TABLE]

By Lemma 2.1, we have $|\varTheta_{\tau}|\tau\ll(|{\mathcal{G}}||\varTheta_{\tau}|)^{2/3}$ provided

[TABLE]

and

[TABLE]

We also define the set

[TABLE]

Comparing (2.8) and (2.11), we see that we can think of $\varTheta_{\tau}$ as of an union of cosets ${\mathcal{Q}}_{\tau}/{\mathcal{G}}$. Clearly, we have

[TABLE]

provided the conditions (2.9) and (2.10) are satisfied.

The condition (2.10) is trivial to verify. Indeed, since $|{\mathcal{G}}|^{2}>p$, we have

[TABLE]

and thus (2.10) holds.

We now show that the condition (2.9) also holds for the following choice

[TABLE]

with a sufficiently large constant $c$ (recalling that $|{\mathcal{G}}|>p^{1/2}$ we see that the condition $\Delta\geq 3$ is satisfied).

Lemma 2.7**.**

For $\Delta$ given by (2.13) the bound (2.9) holds.

Proof.

Suppose, to the contrary, that

[TABLE]

Whence, the number of incidences between points of ${\mathcal{P}}={\mathcal{G}}^{2}$ and the lines ${\mathcal{L}}_{\alpha,\beta}$ as above with $(\alpha,\beta)\in{\mathcal{Q}}_{\tau}$ is at least

[TABLE]

On the other hand, by a classical result which holds over any field (see, for example [7, Corollary 5.2] or [37, Exercise 8.2.1]) the number of incidences for any set of points ${\mathcal{P}}$ and a set of lines ${\mathcal{Q}}_{\tau}$ is at most $|{\mathcal{Q}}_{\tau}|^{1/2}|{\mathcal{P}}|+|{\mathcal{Q}}_{\tau}|$. Hence

[TABLE]

and we obtain

[TABLE]

Combining (2.15) and (2.17), we derive

[TABLE]

Recalling that $|{\mathcal{G}}|\geq p^{1/2}$, we see that for $\Delta$ given by (2.13) with a sufficiently large constant $c$ the inequalities (2.18) are impossible, which also shows that our assumption (2.14) is false and this concludes the proof.     \sqcap$$\sqcup

2.4. Concluding the proof of Theorem 1.2

We now define

[TABLE]

By Lemma 2.7, for the choice (2.13) of $\Delta$ we have the desired condition (2.9) for any $\tau\geq\Delta$. Hence, the bound (2.12) also implies that

[TABLE]

We see from (2.7) that there is a one-to-one correspondence between the lines $\ell_{a,b}$, $(a,b)\in\left({\mathbb{F}}_{p}^{*}\right)^{2}$ and the lines ${\mathcal{L}}_{\alpha,\beta}$, $(\alpha,\beta)\in\left({\mathbb{F}}_{p}^{*}\right)^{2}$. We now define

[TABLE]

where

[TABLE]

Note that due to the choice of $\Delta$ and the condition $|{\mathcal{G}}|\geq p^{1/2}$ we have

[TABLE]

Then, recalling also the bound (2.19), we conclude that the contribution to $W^{*}$ from the lines with $\tau_{j+1}\geq\iota_{\mathcal{G}}\left(\ell_{a,b}\right)>\tau_{j}$ is bounded by

[TABLE]

Summing up (2.20) we obtain

[TABLE]

Substituting this bound in (2.5) and combining it with (2.4), we obtain

[TABLE]

in the range $p^{2/3}\geq|{\mathcal{G}}|\geq p^{1/2}$, which concludes the proof.

Remark 2.8**.**

In principle, a stronger version of the classical incidence bound which is used (2.16) may lead to improvements of Theorem 1.2. However, the range where such improvements are known is far away from the range which appears in our applications, see [35].

3. Trinomial sums

3.1. Preliminaries

We recall the following classical bound of bilinear sums, see, for example, [17, Lemma 4.1].

Lemma 3.1**.**

For any sets ${\mathcal{X}},{\mathcal{Y}}\subseteq{\mathbb{F}}_{p}$ and any $\alpha=(\alpha_{x})_{x\in{\mathcal{X}}}$, $\beta=\left(\beta_{y}\right)_{y\in{\mathcal{Y}}}$, with

[TABLE]

we have

[TABLE]

Definition 3.2** (Ratios of differences).**

For a set ${\mathcal{U}}\subseteq{\mathbb{F}}^{*}_{p}$, we define $D_{\times}({\mathcal{U}})$ to be the number of solutions of

[TABLE]

As before we define ${\mathrm{T}}({\mathcal{U}})$ as the number of solutions to (1.5).

We now recall the following bound from [25, Lemma 2.7].

Lemma 3.3**.**

For any set ${\mathcal{U}}\subseteq{\mathbb{F}}_{p}^{*}$ with $|{\mathcal{U}}|=U$, we have

[TABLE]

Combining Lemma 3.3 with Theorem 1.2 we obtain

[TABLE]

Since for $|{\mathcal{G}}|\geq\left(p\log p\right)^{1/2}$ the first term dominates, this simplifies as

Corollary 3.4**.**

For a multiplicative subgroup ${\mathcal{G}}\subseteq{\mathbb{F}}_{p}^{*}$, we have

[TABLE]

Substituting in Corollary 3.4 into the proof of [25, Theorem 1.3], we obtain the following result for trilinear sums over subgroups, which improves its general bound.

Lemma 3.5**.**

For any multiplicative subgroups ${\mathcal{F}},{\mathcal{G}},{\mathcal{H}}\subseteq{\mathbb{F}}_{p}^{*}$ of cardinalities $F,G,H$, respectively, with $F\geq G\geq H$ and weights $\rho=(\rho_{u,v})$, $\sigma=(\sigma_{u,w})$ and $\tau=(\tau_{v,w})$ with

[TABLE]

for the sum

[TABLE]

we have

[TABLE]

uniformly over $a\in{\mathbb{F}}_{p}^{*}$.

Proof.

We see from [25, Equation (3.8)] that

[TABLE]

where $K$ is the number of solutions to the equation

[TABLE]

As in the proof of [25, Theorem 1.3], expressing $K$ via multiplicative character sums and using the Cauchy inequality, we obtain $K^{2}\leq D_{\times}({\mathcal{G}})D_{\times}({\mathcal{H}})$. Applying Corollary 3.4, instead of [25, Equation 3.9], we now obtain

[TABLE]

We now deal with the three cases separately.

For $H\geq\left(p\log p\right)^{1/2}$ we have

[TABLE]

Since $F<p<H^{2}$, the first term dominates, and we obtain

[TABLE]

For $G\geq\left(p\log p\right)^{1/2}>H$, we have

[TABLE]

or

[TABLE]

The first term of (3.2) dominates for $p^{1/2}\geq F/H$.

We now note that by Lemma 3.1 and the trivial bound for the sum over ${\mathcal{H}}$, we also have

[TABLE]

Furthermore, since for $F>p^{1/2}H$ and $G>p^{1/2}$ we have

[TABLE]

we see that for $G\geq\left(p\log p\right)^{1/2}>H$ the bound (3.2) simplifies as

[TABLE]

For $G<\left(p\log p\right)^{1/2}$, we have

[TABLE]

or

[TABLE]

The first term of (3.5) dominates for $pH\geq FG$. Otherwise, that is, for $pH<FG$, we have

[TABLE]

Thus, using (3.3) we see that the bound (3.5) simplifies as

[TABLE]

Combining (3.1), (3.4) and (3.6), we complete the proof.     \sqcap$$\sqcup

Clearly, the bound of Lemma 3.5 is nontrivial when $F$, $G$ and $H$ are all a little larger than $p^{1/3}$. More formally, for any $\varepsilon>0$ there exists some $\delta>0$ such that if $F\geq G\geq H\geq p^{1/3+\varepsilon}$ then the exponential sums of Lemma 3.5 are bounded by $O\left(FGHp^{-\delta}\right)$.

3.2. Proof of Theorem 1.5

Let ${\mathcal{G}}_{d}$ and ${\mathcal{G}}_{e}$ be the subgroups of ${\mathbb{F}}_{p}^{*}$ formed by the elements of orders dividing $d$ and $e$, respectively.

We have,

[TABLE]

where

[TABLE]

Clearly, the set ${\mathcal{X}}=\{x^{m}~{}:~{}x\in{\mathbb{F}}_{p}^{*}\}$ of non-zero $m$th powers contains $(p-1)/f$ elements, each appearing with multiplicity $f$. Furthermore, direct examination shows that the sets ${\mathcal{Y}}=\{y^{m}~{}:~{}y\in{\mathcal{G}}_{d}\}$ and ${\mathcal{Z}}=\{z^{m}~{}:~{}z\in~{}{\mathcal{G}}_{e}\}$ contain $g$ and $h$ elements with multiplicities $\gcd(d,f)$ and $\gcd(e,f)$, respectively. We recall that by our assumption we have $f\geq g\geq h$ and invoke Lemma 3.5, which gives us,

[TABLE]

This concludes the proof.

4. Further Applications

4.1. Additive properties of subgroups

As usual, given a rational function

[TABLE]

and $m$ sets ${\mathcal{A}}_{1},\ldots,{\mathcal{A}}_{m}\subseteq{\mathbb{F}}_{p}$, we define the set

[TABLE]

where ${\mathcal{P}}_{R}$ is the set of poles of $R$.

We note that we have used ${\mathcal{A}}^{m}$ for the $m$-fold Cartesian product rather than for the $m$-fold product-set of a set ${\mathcal{A}}$ as the previous definition suggests. However neither of these notations is used in this section.

For a scalar $\lambda\in{\mathbb{F}}_{p}$ we use the notation

[TABLE]

for sets of multiples of ${\mathcal{A}}\subseteq{\mathbb{F}}_{p}$.

Applying the bound of Theorem 1.2 to cosets of ${\mathcal{G}}$, that is, to $T({\mathcal{G}},\lambda{\mathcal{G}})$, and using (2.3) we obtain:

Corollary 4.1**.**

Let ${\mathcal{G}}$ be a multiplicative subgroup of ${\mathbb{F}}_{p}^{*}$. Then for any $\lambda\in{\mathbb{F}}_{p}^{*}$, we have

[TABLE]

Note that for $|{\mathcal{G}}|/\sqrt{p\log p}\to\infty$, Corollary 4.1 gives an asymptotic formula for ${\mathrm{E}}^{\times}({\mathcal{G}}+\lambda)$; otherwise we only have an upper bound.

Corollary 4.2**.**

For a multiplicative subgroup ${\mathcal{G}}$ of ${\mathbb{F}}_{p}^{*}$ and $\lambda,\mu\in{\mathbb{F}}_{p}^{*}$ we define the sets

[TABLE]

We have:

• •

if $|{\mathcal{G}}|\geq p^{2/3}$ then ${\mathbb{F}}_{p}^{*}\subseteq{\mathcal{S}}_{1}$ and ${\mathbb{F}}_{p}^{*}\subseteq{\mathcal{G}}{\mathcal{S}}_{2}$;

• •

if $|{\mathcal{G}}|\leq(p\log p)^{1/2}$ then, for $i=1,2$,

[TABLE]

• •

otherwise, for $i=1,2$,

[TABLE]

Proof.

We consider the set ${\mathcal{S}}_{1}$ first.

First we show that ${\mathcal{S}}_{1}\supseteq{\mathbb{F}}^{*}_{p}$, provided $|{\mathcal{S}}_{1}|\geq p^{2/3}$. Clearly, the set ${\mathcal{S}}_{1}$ satsifies the property ${\mathcal{S}}_{1}{\mathcal{G}}={\mathcal{S}}_{1}$ and hence if ${\mathcal{S}}_{1}\supseteq{\mathbb{F}}^{*}_{p}$, then there is a nonzero $\xi$ such that ${\mathcal{S}}_{1}\cap\xi{\mathcal{G}}=\emptyset$. In other words, the equation

[TABLE]

has no solutions. By the orthogonality property of exponential functions, this means that for the sum

[TABLE]

we have

[TABLE]

Clearly, the contribution of $\sigma$ corresponding to $a=0$ equals $|{\mathcal{G}}|^{4}$. Using the well-known bound

[TABLE]

see, for example [19, Equation (1)], combined with the identity

[TABLE]

where $k=(p-1)/|{\mathcal{G}}|$, we obtain

[TABLE]

By the Cauchy inequality, we get

[TABLE]

and this is a contradiction which gives the result for $|{\mathcal{G}}|\geq p^{2/3}$.

We now consider subgroups with $|{\mathcal{G}}|<p^{2/3}$. Clearly

[TABLE]

For $a\in{\mathbb{F}}_{p}^{*}$, we let $N(a)$ be the number of solutions to $(g+\lambda)(h+\mu)=a$ with $g,h\in{\mathcal{G}}$. Clearly

[TABLE]

Hence, by the Cauchy inequality, we have

[TABLE]

where $F$ is the number of solutions to

[TABLE]

There are obviously $O(|{\mathcal{G}}|^{2})$ solutions when

[TABLE]

For the other solutions we repeat the same argument as in the above. That is, for every $a\in{\mathbb{F}}_{p}$, we first collect together solutions with the same value

[TABLE]

After this, using the Cauchy inequality again, we obtain

[TABLE]

Hence, putting the above inequalities together, we derive,

[TABLE]

Hence, using Corollary 4.1, we derive the result for ${\mathcal{S}}_{1}$. Indeed, let $\mathfrak{R}$ be the bound on $\left|{\mathrm{E}}^{\times}({\mathcal{G}}+\lambda)-|{\mathcal{G}}|^{4}/p\right|$ given by Corollary 4.1. It is easy to see that for ${\mathcal{G}}$ to which the upper bound on $p-|{\mathcal{S}}_{i}|$ applies we have

[TABLE]

Hence

[TABLE]

which, together with $\mathfrak{R}\gg|{\mathcal{G}}|^{2}$, implies

[TABLE]

where

[TABLE]

We note that by adjusting the implied constant in the upper bound on $p-|{\mathcal{S}}_{1}|$ we see that one can actually assume that $|{\mathcal{G}}|\geq C_{0}(p\log p)^{1/2}$ for some sufficiently large absolute constant $C_{0}$, so that $|\mathfrak{Q}|\leq 1/2$. In this case

[TABLE]

and the bound on $p-|{\mathcal{S}}_{1}|$ follows. For the lower bound on $|S_{1}|$ we simply remark that the error term $\mathfrak{R}$ dominates the main term $|{\mathcal{G}}|^{4}/p$ in Corollary 4.1, so in this case we simply write

[TABLE]

and the bound follows.

Similar arguments also lead to the same bounds on $|{\mathcal{S}}_{2}|$. For example, consider the case $|{\mathcal{G}}|\geq p^{2/3}$ (where the statement about ${\mathcal{S}}_{2}$ is slightly different than that about ${\mathcal{S}}_{1}$). We denote

[TABLE]

Using the orthogonality of exponential functions, for any $\xi\in{\mathbb{F}}_{p}^{*}$, we can write

[TABLE]

As before, we obtain

[TABLE]

Hence, for $|{\mathcal{G}}|>p^{2/3}$ we have

[TABLE]

Therefore, there is a solution with $v_{1}\neq v_{2}$ which leads to a representation $\xi=(\lambda u_{1}-u_{2})/(v_{1}-v_{2})$ for every $\xi\in{\mathbb{F}}_{p}^{*}$.

Proofs of the other statements about ${\mathcal{S}}_{2}$ are the same as those about ${\mathcal{S}}_{1}$.     \sqcap$$\sqcup

In particular, Corollary 4.2 applies to ${\mathcal{S}}_{1}={\mathcal{G}}+{\mathcal{G}}+{\mathcal{G}}$ and ${\mathcal{S}}_{1}={\mathcal{G}}+{\mathcal{G}}-{\mathcal{G}}$.

We note that for the set ${\mathcal{Q}}$ given by (4.1) we have $0\in\frac{\lambda{\mathcal{G}}-{\mathcal{G}}}{\mu{\mathcal{G}}-{\mathcal{G}}}$ if and only if $\lambda\in{\mathcal{G}}$.

Remark 4.3**.**

Let

[TABLE]

Clearly,

[TABLE]

Hence the set ${\mathcal{Q}}$ contains both ${\mathcal{R}}{\mathcal{G}}$ and $(1-{\mathcal{R}}){\mathcal{G}}$ and hence $|{\mathcal{Q}}|\geq\max\{|{\mathcal{R}}{\mathcal{G}}|,|(1-{\mathcal{R}}){\mathcal{G}}|\}$. Using [30, Theorem 18] and $|{\mathcal{R}}|\gg|{\mathcal{G}}|^{2}/\log|{\mathcal{G}}|$ one can show that there is an absolute constant $c>0$ such that $|{\mathcal{Q}}|\gg|{\mathcal{G}}|^{2+c}$ for sufficiently small ${\mathcal{G}}$ (the condition $|{\mathcal{Q}}|^{2}|{\mathcal{G}}|\leq p^{2}$ is enough). Thus the lower bound for size of ${\mathcal{Q}}$ which follows from bounds on $|{\mathcal{S}}_{2}|$ in Corollary 4.2 can be improved for small subgroups.

We note that Corollary 4.2 also allows us to obtain the following version of the Romanoff theorem modulo almost all primes $p$.

Corollary 4.4**.**

For a fixed integer $g$ with $|g|\geq 2$, and sufficiently large $Q$, for all but $o(Q/\log Q)$ primes $p\leq Q$ every residue class modulo $p$ can be represented as $\ell+g^{k}+g^{m}+g^{n}$ for a prime $\ell<p$ and positive integers $k,m,n\leq p-1$.

Proof.

We recall that by a special case of a result of Indlekofer and Timofeev [20, Corollary 6], given any positive $\alpha<1$, for all but $o(Q/\log Q)$ primes $p\leq Q$, the multiplicative order of $g$ modulo $p$ is at least $p^{1/2}\exp\left((\log p)^{\alpha}\right)$. For each of these primes, we apply Corollary 4.2 to the set ${\mathcal{S}}_{1}={\mathcal{G}}+{\mathcal{G}}+{\mathcal{G}}$ with the group ${\mathcal{G}}\equiv\langle g\pmod{p}\rangle$ (only the first two inequalities are relevant) and use that for $|{\mathcal{G}}|\geq p^{2/3}$ we have $p^{5/2}|{\mathcal{G}}|^{-5/2}\leq p^{3/2}|{\mathcal{G}}|^{-1}$. Hence we obtain

[TABLE]

and by the prime number theorem we conclude the proof.     \sqcap$$\sqcup

We remark that a classical result of Erdős and Murty [15] can also be used in the proof of Corollary 4.4, however the bound of [20, Corollary 6] used in full strength allows to get better estimates on the size of the exceptional set. Perhaps more recent results of Ford [16] can also be used to estimate the size of the exceptional set, however we do not pursue this here.

4.2. Possible application to arbitrary sets

Note that some auxiliary results established in the proofs of [18, Theorems 1 and 2] can be reformulated as bounds on the size of the set $({\mathcal{A}}-{\mathcal{A}})({\mathcal{A}}-{\mathcal{A}})$ for an arbitrary set ${\mathcal{A}}\subseteq{\mathbb{F}}_{p}$. We also refer to [2] for more recent results and references. Combined with the ideas of Balog [2] this may lead to further results on additive properties of the product sets of difference sets.

Acknowledgements

The authors would like to thank Giorgis Petridis for his comments and suggestions and the referee for the very careful reading of the manuscript and numerous corrections.

During the preparation of this work, the third author was supported by the Australian Research Council Grant DP170100786.