We give a short proof of the fact that there are no measurable subsets of Euclidean space (in dimension $d\ge 3$) which, no matter how translated and rotated, always contain exactly one integer lattice point. In dimension $d=2$ (the original Steinhaus problem) the question remains open.
"The Steinhaus tiling problem and the range of certain quadratic forms." Illinois J. Math. 46 (3) 947 - 951, Fall 2002. https://doi.org/10.1215/ijm/1258130994