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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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 $ {\mathbb{F}}_q$, the finite field with $ q$ elements, by $ A \cdot A+\dots +A \cdot A$, where $ A$ is a subset $ {\mathbb{F}}_q$ 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 $ \mathbb{F}_q^d$ 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: http://dx.doi.org/10.1090/S0002-9947-2010-05232-8
PII: S 0002-9947(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.