Slightly improved sum-product estimates in fields of prime order

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 $|A|⪯p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succeq|A|^{13/12};\] if $|A|⪰p^{0.5}$, 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.