On the Steinhaus tiling problem for \(\mathbb{Z}^3\) (Q2922426)
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scientific article; zbMATH DE number 6353640
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the Steinhaus tiling problem for \(\mathbb{Z}^3\) |
scientific article; zbMATH DE number 6353640 |
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10 October 2014
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Steinhaus problem
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tiling
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Steinhaus set
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three-dimensional
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heuristic evidence
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0.9012795
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0.89169943
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0.88065976
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0.8719874
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0.86746776
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0.86719984
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0.8639182
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On the Steinhaus tiling problem for \(\mathbb{Z}^3\) (English)
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In 1957 H.~Steinhaus asked if there is a set in \(\mathbb{R}^2\) that meets every isometric copy of \(\mathbb{Z}^2\) (integer lattice) in exactly one point. Such a `Steinhaus set' was constructed by \textit{S. Jackson} and \textit{R. D. Mauldin} [J. Am. Math. Soc. 15, No. 4, 817--856 (2002; Zbl 1021.03040)].NEWLINENEWLINEThe paper under review discusses the analogous three-dimensional question: is there a set in \(\mathbb{R}^3\) that meets every isometric copy of \(\mathbb{Z}^3\) in exactly one point? The authors offer heuristic evidence that the answer is negative.
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