A sum-product estimate in fields of prime order

TL;DR

This paper proves a sum-product estimate in finite fields of prime order, showing that either the sum set or the product set of a subset must be significantly larger than the original set, extending previous results.

Contribution

It establishes a new lower bound for the maximum of sum and product set sizes in prime fields, advancing the understanding of sum-product phenomena.

Findings

Either |A+A| or |A·A| is at least c|A|^{1+ε} for some ε>0 and c>0

Extends Bourgain, Katz, Tao's sum-product estimate to broader settings

Provides quantitative bounds for sum-product growth in finite fields

Abstract

Let q be a prime, A be a subset of a finite field $F=\Bbb Z/q\Bbb Z$, $|A|<\sqrt{|F|}$. We prove the estimate $\max(|A+A|,|A\cdot A|)\ge c|A|^{1+\epsilon}$ for some $\epsilon>0$ and c>0. This extends the result of J. Bourgain, N. Katz, and T. Tao.