A sum-product estimate in fields of prime order

Abstract

Let q be a prime, A be a subset of a finite field $F=\mathbb{Z}/q\mathbb{Z}$, $|A|<\sqrt{|F|}$. 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.