Character sums estimates and an application to a problem of Balog

TL;DR

This paper establishes new bounds for character sums over sets with small doubling and applies these results to solve a problem of Balog, surpassing the square-root barrier in prime fields.

Contribution

It introduces novel bounds for multiplicative character sums over small doubling sets and applies them to a longstanding problem of Balog in prime fields.

Findings

New bounds for character sums over small doubling sets

Breakthrough in Balog's problem surpassing the square-root barrier

Enhanced understanding of product of differences in prime fields

Abstract

We prove new bounds for sums of multiplicative characters over sums of set with small doubling and applying this result we break the square--root barrier in a problem of Balog concerning products of differences in a field of prime order.