Comments about the Steinhaus tiling problem

Recently, using Fourier transform methods, it was shown that there is no measurable Steinhaus set in $\mathbb{R} ^{3}$, a set which no matter how translated and rotated contains exactly one integer lattice point. Here, we show that this argument cannot generalize to any lattice and, on the other hand, give some lattices to which this method applies. We also show there is no measurable Steinhaus set for a special honeycomb lattice, the standard tetrahedral lattice in $\mathbb{R} ^3.$