Distribution of residues in approximate subgroups of $\mathbb{F}_p^*$

TL;DR

This paper proves that certain structured subsets of finite fields, formed by polynomial images of intervals multiplied by approximate subgroups, are uniformly distributed as the field size grows large.

Contribution

It extends Bourgain's result by establishing equidistribution for subsets involving polynomial images and approximate subgroups in finite fields.

Findings

Subsets of the form f(I)·H are equidistributed in _p as p rrows ty.

The result applies to H larger than polylogarithmic in p.

Generalizes previous uniform distribution results to new structured sets.

Abstract

We extend a result due to Bourgain on the uniform distribution of residues by proving that subsets of the type $f(I)\cdot H$ is equidistributed (as $p$ tends to infinity) where $f$ is a polynomial, $I$ is an interval of $\Fp$ and $H$ is an approximate subgroup of $\mathbb{F}_p^*$ with size larger than polylogarithmic in $p$.