# A sum-product estimate in finite fields, and applications

## Authors

- Received:
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DOI: 10.1007/s00039-004-0451-1

- Cite this article as:
- Bourgain, J., Katz, N. & Tao, T. Geom. funct. anal. (2004) 14: 27. doi:10.1007/s00039-004-0451-1

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## 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.