Additive double character sums over some structured sets and applications

TL;DR

This paper investigates additive double character sums over structured subsets of finite fields, introducing new bounds when the sets admit certain rational self-maps, and applies these results to trace products and sum-product equations.

Contribution

It establishes improved bounds on additive double character sums for sets with rational self-maps, enhancing understanding of sum-product phenomena in finite fields.

Findings

Improved bounds on character sums for sets with rational self-maps.

Applications to trace products and sum-product equations.

Enhanced results over previous work by the authors and others.

Abstract

We study additive double character sums over two subsets of a finite field. We show that if there is a suitable rational self-map of small degree of a set $D$, then this set contains a large subset $U$ for which the standard bound on the absolute value of the character sum over $U$ and any subset $C$ (which satisfies some restrictions on its size $|C|$) can be improved. Examples of such suitable self-maps are inversion and squaring. Then we apply this new bound to trace products and sum-product equations and improve results of the first author and of Gyarmati and S\'ark\"ozy.