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 $\mathbb {F}_p$. In the proofs we use the usual incidence theorems in $\mathbb {F}_p$, as well as the growth result in $\mathrm {SL}_2 (\mathbb {F}_p)$ due to Helfgott. Here are 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 $\mathbb {F}_p$, $\bullet ~$ 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 $\mathbb {F}_p$, $p_1,p_2 \in \mathbb {F}_p [x]$ are two non–constant polynomials, $\bullet ~$ new bounds for some exponential sums with multiplicative and additive characters.