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Steinhaus tiling problem and integral quadratic forms

Authors: Wai Kiu Chan and R. Daniel Mauldin
Journal: Proc. Amer. Math. Soc. 135 (2007), 337-342
MSC (2000): Primary 11E12, 11H06, 28A20
Published electronically: August 4, 2006
MathSciNet review: 2255279
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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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Additional Information

Wai Kiu Chan
Affiliation: Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut 06459

R. Daniel Mauldin
Affiliation: Department of Mathematics, Box 311430, University of North Texas, Denton, Texas 76203

Keywords: Representations by quadratic forms, Steinhaus tiling problem
Received by editor(s): August 8, 2005
Received by editor(s) in revised form: August 29, 2005
Published electronically: August 4, 2006
Additional Notes: The research of the first author was partially supported by NSF grant DMS-0138524
The second author was supported in part by NSF grant DMS-0400481
Communicated by: Ken Ono
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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