Erdös distance problem in vector spaces over finite fields
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- by A. Iosevich and M. Rudnev PDF
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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.References
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Additional Information
- A. Iosevich
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 356191
- 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
- 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.
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 6127-6142
- MSC (2000): Primary 11T24, 52C10
- DOI: https://doi.org/10.1090/S0002-9947-07-04265-1
- MathSciNet review: 2336319