Partial Gaussian sums in finite fields

TL;DR

This paper extends Burgess's partial Gaussian sum results to all finite fields using advanced combinatorial and geometric methods, improving understanding of character sums in finite fields.

Contribution

It introduces a generalized approach to partial Gaussian sums in finite fields, incorporating new techniques from additive combinatorics and geometry of numbers.

Findings

Generalization of Burgess' results to arbitrary finite fields

Incorporation of deep results on multiplicative energy in finite fields

Enhanced bounds on character sums in finite fields

Abstract

We generalize Burgess' results on partial Gaussian sums to arbitrary finite fields. The main ingredients are the classical method of amplification, two deep results on multiplicative energy for subsets in finite fields which are obtained respectively by the tools from additive combinatorics and geometry of numbers, and a technique of Chamizo for treating the difficulty caused by additive character. Our results include the recent works on character sums in finite fields by M.-C. Chang and S. V. Konyagin.