On asymptotic formulae in some sum-product questions

TL;DR

This paper develops asymptotic formulae for sum-product problems over prime fields using incidence theorems and growth results, leading to new bounds and analogues for equations, exponential sums, and decompositions.

Contribution

It introduces novel asymptotic formulae and bounds in sum-product phenomena over prime fields, extending existing results with new applications and analogues.

Findings

New bound for solutions to a specific product-sum equation

Effective bounds for multilinear exponential sums

Asymptotic analogue of the Balog–Wooley decomposition

Abstract

In this paper we obtain a series of asymptotic formulae in the sum--product phenomena over the prime field $\mathbf{F}_p$. In the proofs we use usual incidence theorems in $\mathbf{F}_p$, as well as the growth result in ${\rm SL}_2 (\mathbf{F}_p)$ due to Helfgott. Here some of our applications: $\bullet~$ a new bound for the number of the solutions to the equation $(a_1-a_2) (a_3-a_4) = (a'_1-a'_2) (a'_3-a'_4)$, $\,a_i, a'_i\in A$, $A$ is an arbitrary subset of $\mathbf{F}_p$, $\bullet~$ a new effective bound for multilinear exponential sums of Bourgain, $\bullet~$ an asymptotic analogue of the Balog--Wooley decomposition theorem, $\bullet~$ growth of $p_1(b) + 1/(a+p_2 (b))$, where $a,b$ runs over two subsets of $\mathbf{F}_p$, $p_1,p_2 \in \mathbf{F}_p [x]$ are two non--constant polynomials, $\bullet~$ new bounds for some exponential sums with multiplicative and additive characters.