Fernando Xuancheng Shao
University of Kentucky
Title
On the sum-product phenomenon in finite fields
Abstract
The sum–product problem asks how small the sumset A+A and product set A*A can simultaneously be. Arithmetic progressions have small sumsets, while geometric progressions have small product sets; the challenge is to understand how much additive and multiplicative structure can coexist. The Erdős–Szemerédi conjecture predicts that for finite sets of integers, at least one of these sets has nearly quadratic size. I will introduce this circle of problems and discuss recent work for dense subsets of finite fields. A central tool is an arithmetic regularity lemma for finite abelian groups.
