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ISSN 1088-6850(e) ISSN 0002-9947(p)

Sets with integral distances in finite fields

Author(s): Alex Iosevich; Igor E. Shparlinski; Maosheng Xiong
Journal: Trans. Amer. Math. Soc. 362 (2010), 2189-2204.
MSC (2000): Primary 05B25, 11T23, 52C10
Posted: November 17, 2009
MathSciNet review: 2574892
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Abstract: Given a positive integer , a finite field $\mathbb{F}_q$ of elements ( odd), and a non-degenerate quadratic form 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 -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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