Steinhaus tiling problem and integral quadratic forms

Chan, Wai Kiu; Mauldin, R.

doi:10.1090/S0002-9939-06-08479-6

Steinhaus tiling problem and integral quadratic forms
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by Wai Kiu Chan and R. Daniel Mauldin PDF

Proc. Amer. Math. Soc. 135 (2007), 337-342 Request permission

Abstract:

A lattice $L$ in $\mathbb {R}^n$ is said to be equivalent to an integral lattice if there exists a real number $r$ such that the dot product of any pair of vectors in $rL$ is an integer. We show that if $n \geq 3$ and $L$ is equivalent to an integral lattice, then there is no measurable Steinhaus set for $L$, a set which no matter how translated and rotated contains exactly one vector in $L$.

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