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

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 ${\mathbb{F}}_p^{\ast }$ with size larger than polylogarithmic in $p$.