Quantitative Sum Product Estimates on Different Sets

Abstract

Let $F_p$ be a finite field of $p$ elements with $p$ prime. In this paper we show that for $A ,B \subset F_p$ with $|B|\leq |A| < p^{{1 \over 2}}$ then $$\max\big(|A+B|, |AB|\big) \gtrapprox \bigg({|B|^{14} \over |A|^{13}}\bigg)^{1/18}|A|.$$ This gives an explicit exponent in a sum-product estimate for different sets by Bourgain.