A sum-product estimate in finite fields, and applications

Abstract

Let A be a subset of a finite field $F := \mathbf{Z}/q\mathbf{Z}$ for some prime q. If $|F|^{\delta} < |A| < |F|^{1-\delta}$ for some δ > 0, then we prove the estimate $|A + A| + |A \cdot A| \geq c(\delta)|A|^{1+\varepsilon}$ for some ε = ε(δ) > 0. This is a finite field analogue of a result of [ErS]. We then use this estimate to prove a Szemerédi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdös distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.