On the Steinhaus tiling problem in three dimensions
Abstract
H. Steinhaus asked in the 1950's whether there exists a set in the plane R2 meeting every isometric copy of Z2 in precisely one point. Such a "Steinhaus set" was constructed by Jackson and Mauldin. What about three-space R3? Is there a subset of R3 meeting every isometric copy of Z3 in exactly one point? We offer heuristic evidence that the answer is "no".
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