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

 

Sets with integral distances in finite fields


Authors: Alex Iosevich, Igor E. Shparlinski and Maosheng Xiong
Journal: Trans. Amer. Math. Soc. 362 (2010), 2189-2204
MSC (2000): Primary 05B25, 11T23, 52C10
Published electronically: November 17, 2009
MathSciNet review: 2574892
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Abstract: Given a positive integer $ n$, a finite field $ \mathbb{F}_q$ of $ q$ elements ($ q$ odd), and a non-degenerate quadratic form $ Q$ on $ \mathbb{F}_q^n$, in this paper we study the largest possible cardinality of subsets $ \mathcal{E} \subseteq \mathbb{F}_q^n$ with pairwise integral $ Q$-distances; that is, for any two vectors $ {\bf {x}}=(x_1, \ldots,x_n), {\bf {y}}=(y_1,\ldots,y_n) \in \mathcal{E}$, one has

$\displaystyle Q({\bf {x}}-{\bf {y}})=u^2$

for some $ u \in \mathbb{F}_q$.


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Additional Information

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

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Email: igor@ics.mq.edu.au

Maosheng Xiong
Affiliation: Department of Mathematics, Eberly College of Science, Pennsylvania State University, State College, Pennsylvania 16802
Email: xiong@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-05004-1
PII: S 0002-9947(09)05004-1
Keywords: Integral distances, Gauss sums
Received by editor(s): September 10, 2008
Published electronically: November 17, 2009
Article copyright: © Copyright 2009 American Mathematical Society