%0 Journal Article
%T A sum-product estimate in finite fields, and applications
%A Jean Bourgain
%A Nets Katz
%A Terence Tao
%J Mathematics
%D 2003
%I arXiv
%X Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If $|F|^\delta < |A| < |F|^{1-\delta}$ for some $\delta > 0$, then we prove the estimate $|A+A| + |A.A| \geq c(\delta) |A|^{1+\eps}$ for some $\eps = \eps(\delta) > 0$. This is a finite field analogue of a result of Erdos and Szemeredi. We then use this estimate to prove a Szemeredi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdos distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.
%U http://arxiv.org/abs/math/0301343v3