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

@article{Bourgain2003ASE,
title={A sum-product estimate in finite fields, and applications},
author={J. Bourgain and N. Katz and T. Tao},
journal={Geometric & Functional Analysis GAFA},
year={2003},
volume={14},
pages={27-57}
}
• Published 2003
• Mathematics
• Geometric & Functional Analysis GAFA
• AbstractLet 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… CONTINUE READING
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