Slightly improved sum-product estimates in fields of prime order

TL;DR

This paper improves sum-product estimates in finite fields of prime order, providing tighter bounds for the size of sum and product sets of subsets of the field, especially when the subset size varies relative to the field size.

Contribution

It introduces refined bounds for sum-product estimates in prime fields, improving upon previous results by Bourgain-Garaev and Shen, especially for different subset sizes.

Findings

For small subsets, max of sum and product sets is at least |A|^{13/12}.

For larger subsets, bounds depend on the ratio of |A| to p^{0.5}.

Results slightly improve existing sum-product estimates in finite fields.

Abstract

Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a nonempty subset of $\mathbb{F}_p$. In this paper we show that if $|A|\preceq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succeq|A|^{13/12};\] if $|A|\succeq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succapprox \min\{|A|^{13/12}(\frac{|A|}{p^{0.5}})^{1/12},|A|(\frac{p}{|A|})^{1/11}\}.\] These results slightly improve the estimates of Bourgain-Garaev and Shen. Sum-product estimates on different sets are also considered.