Steinhaus tiling problem and integral quadratic forms

Abstract

A lattice L L in R n \mathbb {R}^n is said to be equivalent to an integral lattice if there exists a real number r r such that the dot product of any pair of vectors in r L rL is an integer. We show that if n ≥ 3 n \geq 3 and L L is equivalent to an integral lattice, then there is no measurable Steinhaus set for L L , a set which no matter how translated and rotated contains exactly one vector in L L .