A sum-product estimate in finite fields, and applications

Abstract

Let $A$ be a subset of a finite field $F:=\mathbb{Z}/q\mathbb{Z}$ for some prime $q$. If $|F{|}^{\delta }<|A|<|F{|}^{1-\delta }$ for some $\delta >0$, then we prove the estimate $|A+A| + |A.A| \geq c(δ) |A|^{1+\eps}$ for some $\eps = \eps(δ) > 0$. 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.

29 pages. The distance set result needs to be restricted to the case when -1 is not a square