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Elementary methods for incidence problems in finite fields

Volume 177 / 2017

Javier Cilleruelo, Alex Iosevich, Ben Lund, Oliver Roche-Newton, Misha Rudnev Acta Arithmetica 177 (2017), 133-142 MSC: Primary 52C10. DOI: 10.4064/aa8225-10-2016 Published online: 22 December 2016

Abstract

We use elementary methods to prove an incidence theorem for points and spheres in $\mathbb F_q^n$. As an application, we show that any point set $P\subset \mathbb F_q^2$ with $|P|\geq 5q$ determines a positive proportion of all circles. The latter result is an analogue of Beck’s Theorem for circles which is optimal up to multiplicative constants.

Authors

  • Javier CillerueloInstituto de Ciencias Matemáticas
    (CSIC-UAM-UC3M-UCM)
    and
    Departamento de Matemáticas
    Universidad Autónoma de Madrid
    28049 Madrid, Spain
  • Alex IosevichDepartment of Mathematics
    University of Rochester
    Rochester, NY 14627, U.S.A.
    e-mail
  • Ben LundDepartment of Computer Science
    Rutgers, The State University of New Jersey
    Piscataway, NJ 08854, USA
    e-mail
  • Oliver Roche-NewtonInstitute for Financial Mathematics
    and Applied Number Theory
    Johannes Kepler Universität
    4040 Linz, Austria
    e-mail
  • Misha RudnevSchool of Mathematics
    University of Bristol
    University Walk
    Bristol, UK, BS8 1TW
    e-mail

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