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

 

Erdös distance problem in vector spaces over finite fields


Authors: A. Iosevich and M. Rudnev
Journal: Trans. Amer. Math. Soc. 359 (2007), 6127-6142
MSC (2000): Primary 11T24, 52C10
Published electronically: July 20, 2007
MathSciNet review: 2336319
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Abstract: We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let $ {\mathbb{F}}_q$ be a finite field with $ q$ elements and take $ E \subset {\mathbb{F}}^d_q$, $ d \ge 2$. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in $ {\mathbb{F}}^d_q$ to provide estimates for minimum cardinality of the distance set $ \Delta(E)$ in terms of the cardinality of $ E$. Bounds for Gauss and Kloosterman sums play an important role in the proof.


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Additional Information

A. Iosevich
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: iosevich@math.missouri.edu

M. Rudnev
Affiliation: Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Email: m.rudnev@bris.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04265-1
PII: S 0002-9947(07)04265-1
Received by editor(s): September 12, 2005
Received by editor(s) in revised form: January 18, 2006
Published electronically: July 20, 2007
Additional Notes: The work was partly supported by the grant DMS02-45369 from the National Science Foundation, the National Science Foundation Focused Research Grant DMS04-56306, and the EPSRC grant GR/S13682/01.
Article copyright: © Copyright 2007 American Mathematical Society