The Steinhaus tiling problem and the range of certain quadratic forms, Illinois 46

Dedicated to the memory of Tom Wolff We give a short proof of the fact that there are no measurable subsets of Euclidean space (in dimension d ≥ 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. §0. The Steinhaus problem Steinhaus (1957) asked if there exists a subset of the plane which, no matter how translated and rotated, always contains exactly one point with integer coordinates. This question remains unanswered. In this paper we deal only with the measurable version of the Steinhaus problem in dimension d ≥ 2, in which a measurable subset E of Rd is sought with the property that for almost every x ∈ Rd and for almost every isometry S: Rd → Rd∣∣∣(SE + x) ∩ Zd∣∣ ∣ = 1. The Steinhaus property may also be written as follows. For almost all isometries S∑ n∈Z