A sum-product estimate in finite fields, and applications

@article{Bourgain2003ASE,
  title={A sum-product estimate in finite fields, and applications},
  author={Jean Bourgain and Nets Hawk Katz and Terence Tao},
  journal={Geometric \& Functional Analysis GAFA},
  year={2003},
  volume={14},
  pages={27-57}
}
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… 
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  • J. Bourgain, M. Garaev
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2009
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