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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Email: wkchan@wesleyan.edu

R. Daniel Mauldin
Affiliation: Department of Mathematics, Box 311430, University of North Texas, Denton, Texas 76203
Email: mauldin@unt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08479-6
PII: S 0002-9939(06)08479-6
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.