A Quantified Version of Bourgain's Sum-Product Estimate in ${\Bbb F}_p$ for Subsets of Incomparable Sizes

Abstract

Let ${\Bbb F}_p$ be the field of residue classes modulo a prime number $p.$ In this paper we prove that if $A,B\subset {\Bbb F}_p^*,$ then for any fixed $\varepsilon>0,$ $$ |A+A|+|AB|\gg \Bigl(\min\Bigl\{|B|,\, {p\over|A|}\Bigr\}\Bigr)^{1/25-\varepsilon}|A|. $$ This quantifies Bourgain's recent sum-product estimate.