An explicit sum-product estimate in ${\mathbb{F}}_p$

Abstract

Let ${\mathbb{F}}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a non-empty subset of ${\mathbb{F}}_p.$ In this paper we give an explicit version of the sum-product estimate of Bourgain, Katz, Tao and Bourgain, Glibichuk, Konyagin on the size of $\max \{|A+A|,|AA|\}.$ In particular, our result implies that if $1<|A|\le p^{7/13}(\log p{)}^{-4/13},$ then $$\max \{|A+A|,|AA|\}\gg \frac{|A{|}^{15/14}}{(\log |A|{)}^{2/7}}.$$

11 pages