A sum-product estimate in finite fields, and applications

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