Sets with integral distances in finite fields

Alex Iosevich; Igor E. Shparlinski; Maosheng Xiong

doi:10.1090/S0002-9947-09-05004-1

Sets with integral distances in finite fields

Alex Iosevich^*, Igor E. Shparlinski, Maosheng Xiong

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

12 Citations (Scopus)

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Abstract

Given a positive integer n, a finite field F_q of q elements (q odd), and a non-degenerate quadratic form Q on F_qⁿ, in this paper we study the largest possible cardinality of subsets ε⊆ F_qⁿ with pairwise integral Q-dis- tances; that is, for any two vectors × = (xi,⋯,x_n),y = (yi,.,yn) ∞εA,one has Q(x - y) = u² for some u ∞ F _q.

Original language	English
Pages (from-to)	2189-2204
Number of pages	16
Journal	Transactions of the American Mathematical Society
Volume	362
Issue number	4
DOIs	https://doi.org/10.1090/S0002-9947-09-05004-1
Publication status	Published - Apr 2010

Bibliographical note

Copyright 2009 American Mathematical Society. First published in Transactions of the American Mathematical Society, 362(4), 2189-2204, published by the American Mathematical Society. The original article can be found at http://dx.doi.org/10.1090/S0002-9947-09-05004-1

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title = "Sets with integral distances in finite fields",

abstract = "Given a positive integer n, a finite field Fq of q elements (q odd), and a non-degenerate quadratic form Q on Fqn, in this paper we study the largest possible cardinality of subsets ε⊆ Fqn with pairwise integral Q-dis- tances; that is, for any two vectors × = (xi,⋯,xn),y = (yi,.,yn) ∞εA,one has Q(x - y) = u2 for some u ∞ F q.",

author = "Alex Iosevich and Shparlinski, \{Igor E.\} and Maosheng Xiong",

note = "Copyright 2009 American Mathematical Society. First published in Transactions of the American Mathematical Society, 362(4), 2189-2204, published by the American Mathematical Society. The original article can be found at http://dx.doi.org/10.1090/S0002-9947-09-05004-1",

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AU - Shparlinski, Igor E.

AU - Xiong, Maosheng

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N2 - Given a positive integer n, a finite field Fq of q elements (q odd), and a non-degenerate quadratic form Q on Fqn, in this paper we study the largest possible cardinality of subsets ε⊆ Fqn with pairwise integral Q-dis- tances; that is, for any two vectors × = (xi,⋯,xn),y = (yi,.,yn) ∞εA,one has Q(x - y) = u2 for some u ∞ F q.

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JF - Transactions of the American Mathematical Society

IS - 4

ER -

Sets with integral distances in finite fields

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