Comments about the Steinhaus tiling problem
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- by R. Daniel Mauldin and Andrew Q. Yingst PDF
- Proc. Amer. Math. Soc. 131 (2003), 2071-2079 Request permission
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 $\mathbb {R}^3.$References
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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
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 2071-2079
- MSC (2000): Primary 28A20; Secondary 11K31, 60F20
- DOI: https://doi.org/10.1090/S0002-9939-03-07089-8
- MathSciNet review: 1963752