ON THE STEINHAUS TILING PROBLEM FOR ℤ3

Steinhaus asked in the 1950s whether there exists a set in ℝ2 meeting every isometric copy of ℤ2 in precisely one point. Such a ‘Steinhaus set’ was constructed by Jackson and Mauldin. What about ℝ3? Is there a subset S of ℝ3 meeting every isometric copy of ℤ3 in exactly one point? We offer heuristic evidence that the answer is ‘no’.