On multiplicative energy of subsets of varieties

TL;DR

This paper establishes a significant upper bound on the multiplicative energy of large subsets within algebraic varieties in finite groups, with applications to growth, exponential sums, and restriction phenomena.

Contribution

It introduces a novel upper bound for multiplicative energy in algebraic varieties, advancing understanding of subset behavior in finite algebraic groups.

Findings

Derived a non-trivial upper bound for multiplicative energy.

Applied results to growth of conjugacy classes.

Provided estimates for exponential sums and restriction phenomena.

Abstract

We obtain a non--trivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to growth of conjugates classes, estimates of exponential sums and the restriction phenomenon.