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arXiv · 1304.8047

On the Steinhaus tiling problem in three dimensions

Daniel Goldstein·R. Daniel Mauldin
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Abstract

H. Steinhaus asked in the 1950's whether there exists a set in the plane R^2 meeting every isometric copy of Z^2 in precisely one point. Such a "Steinhaus set" was constructed by Jackson and Mauldin. What about three-space R^3? Is there a subset of R^3 meeting every isometric copy of Z^3 in exactly one point? We offer heuristic evidence that the answer is "no".

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Daniel Goldstein, R. Daniel Mauldin. 2013-04-30. On the Steinhaus tiling problem in three dimensions. https://arxiv.org/abs/1304.8047

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