A sum-product estimate in finite fields, and applications

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