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}
}
  • J. Bourgain, N. Katz, T. Tao
  • 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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    References

    SHOWING 1-10 OF 40 REFERENCES
    Structure Theory of Set Addition
    • 34
    • PDF
    On the number of sums and products
    • 135
    • Highly Influential
    • PDF
    A polynomial bound in Freiman's theorem
    • 185
    • PDF
    On sums and products of integers
    • 53
    • PDF
    An analog of Freiman's theorem in groups
    • 104
    Extremal problems in discrete geometry
    • 512
    • PDF
    An improved bound on the Minkowski dimension of Besicovitch sets in $\mathbb{R}^3$
    • 67
    • PDF