Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture

Authors:
Derrick Hart, Alex Iosevich, Doowon Koh and Misha Rudnev

Journal:
Trans. Amer. Math. Soc. **363** (2011), 3255-3275

MSC (2010):
Primary 42B05, 11T23, 52C10

DOI:
https://doi.org/10.1090/S0002-9947-2010-05232-8

Published electronically:
December 29, 2010

MathSciNet review:
2775806

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Abstract: We prove a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering , the finite field with elements, by , where is a subset of sufficiently large size. We also use the incidence machinery and develop arithmetic constructions to study the Erdős-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdős-Falconer distance conjecture does not hold in this setting. On the positive side, we obtain good exponents for the Erdős-Falconer distance problem for subsets of the unit sphere in and discuss their sharpness. This results in a reasonably complete description of the Erdős-Falconer distance problem in higher-dimensional vector spaces over general finite fields.

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Additional Information

**Derrick Hart**

Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019

**Alex Iosevich**

Affiliation:
Department of Mathematics, University of Rochester, Hylan 909, Rochester, New York 14627

**Doowon Koh**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

**Misha Rudnev**

Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW England

DOI:
https://doi.org/10.1090/S0002-9947-2010-05232-8

Received by editor(s):
May 28, 2008

Received by editor(s) in revised form:
September 26, 2009

Published electronically:
December 29, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.