Estimates for Character Sums with Various Convolutions

TL;DR

This paper derives bounds for complex character sums involving convolutions over subsets of finite fields, advancing understanding of their behavior in additive combinatorics and number theory.

Contribution

It introduces new estimates for multi-variable character sums with convolutions, extending previous bounds to more complex sum structures in finite fields.

Findings

Derived bounds for sums over three sets involving (a+b+c)

Established estimates for four-set sums with (a+b+cd)

Enhanced understanding of character sum behavior in finite fields

Abstract

We provide estimates for sums of the form \[\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\chi(a+b+c)\right|\] and \[\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\sum_{d\in D}\chi(a+b+cd)\right|\] when $A,B,C,D\subset \mathbb F_p$, the field with $p$ elements and $\chi$ is a non-trivial multiplicative character modulo $p$.