Sets with integral distances in finite fields
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- by Alex Iosevich, Igor E. Shparlinski and Maosheng Xiong PDF
- Trans. Amer. Math. Soc. 362 (2010), 2189-2204 Request permission
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 $\textbf {{x}}=(x_1, \ldots ,x_n), \textbf {{y}}=(y_1,\ldots ,y_n) \in \mathcal {E}$, one has \[ Q(\textbf {{x}}-\textbf {{y}})=u^2\] for some $u \in \mathbb F_q$.References
Additional Information
- Alex Iosevich
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 356191
- Email: iosevich@math.missouri.edu
- Igor E. Shparlinski
- Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
- MR Author ID: 192194
- 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
- Received by editor(s): September 10, 2008
- Published electronically: November 17, 2009
- © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 362 (2010), 2189-2204
- MSC (2000): Primary 05B25, 11T23, 52C10
- DOI: https://doi.org/10.1090/S0002-9947-09-05004-1
- MathSciNet review: 2574892