Geometric & Functional Analysis GAFA

, Volume 14, Issue 1, pp 27–57

A sum-product estimate in finite fields, and applications

Original Article

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


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.


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Mathematics Subject Classification (2000).

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Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.School of MathematicsInstitute of Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsWashington University in St. LouisSt. LouisUSA
  3. 3.Department of MathematicsUCLALos AngelesUSA