A sum-product estimate in finite fields, and applications

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.

Authors and Affiliations

1. School of Mathematics, Institute of Advanced Study, Princeton, NJ, 08540, USA

   Jean Bourgain

2. Department of Mathematics, Washington University in St. Louis, St. Louis, MO, 63130, USA

   Nets Katz

3. Department of Mathematics, UCLA, Los Angeles, CA, 90095-1555, USA

   Terence Tao

Corresponding authors

Correspondence to Jean Bourgain, Nets Katz or Terence Tao.

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