On Some Multiple Character Sums

TL;DR

This paper improves bounds on certain multiplicative character sums over finite fields, using recent advances in additive combinatorics, and explores sums with specific algebraic structures involving four variables.

Contribution

It provides tighter bounds on multiplicative character sums involving expressions like a + b + cd and a + b(c+d), extending previous results with new combinatorial techniques.

Findings

Enhanced bounds on character sums with algebraic expressions

Application of additive combinatorics to finite field sums

Improved understanding of sum structures in finite fields

Abstract

We improve a recent result of B. Hanson (2015) on multiplicative character sums with expressions of the type $a + b +cd$ and variables $a,b,c,d$ from four distinct sets of a finite field. We also consider similar sums with $a + b(c+d)$. These bounds rely on some recent advances in additive combinatorics.