We survey some results and problems arising from a classic problem of Steinhaus: Is there a subset S of ℝ2 such that each isometric copy of ℤ2 (the lattice points in the plane) meets S in exactly one point.
(Received June 22 2003)
(Revised June 25 2003)
* Research supported by NSF Grant DMS-0097181
† Research supported by NSF Grant DMS-0100078