, Volume 14, Issue 1, pp 27-57

A sum-product estimate in finite fields, and applications

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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.