On asymptotic formulae in some sum-product questions

I. D. Shkredov

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 79 (2018) no. 2, pp. 271-334

Résumé

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: 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$, a new effective bound for multilinear exponential sums of Bourgain, an asymptotic analogue of the Balog–Wooley decomposition theorem, 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, new bounds for some exponential sums with multiplicative and additive characters.