Additive double character sums over some structured sets and applications
Cathy Swaenepoel, Arne Winterhof

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 , then this set contains a large subset for which the standard bound on the absolute value of the character sum over and any subset (which satisfies some restrictions on its size ) 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.
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