Steinhaus tiling problem and integral quadratic forms

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