On asymptotic formulae in some sum–product questions

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.