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

 

Comments about the Steinhaus tiling problem


Authors: R. Daniel Mauldin and Andrew Q. Yingst
Journal: Proc. Amer. Math. Soc. 131 (2003), 2071-2079
MSC (2000): Primary 28A20; Secondary 11K31, 60F20
Published electronically: February 13, 2003
MathSciNet review: 1963752
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Abstract: Recently, using Fourier transform methods, it was shown that there is no measurable Steinhaus set in $\mathbb{R} ^{3}$, a set which no matter how translated and rotated contains exactly one integer lattice point. Here, we show that this argument cannot generalize to any lattice and, on the other hand, give some lattices to which this method applies. We also show there is no measurable Steinhaus set for a special honeycomb lattice, the standard tetrahedral lattice in $\mathbb{R} ^3.$


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Additional Information

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

Andrew Q. Yingst
Affiliation: Department of Mathematics, Box 311430, University of North Texas, Denton, Texas 76203
Email: andyq@unt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07089-8
PII: S 0002-9939(03)07089-8
Keywords: Method of descent, Fourier transform, lattice, quadratic form
Received by editor(s): October 15, 2001
Published electronically: February 13, 2003
Additional Notes: Both authors were supported in part by NSF Grant DMS 0100078
Communicated by: David Preiss
Article copyright: © Copyright 2003 American Mathematical Society