Tiling Lattices with Sublattices, I

Call a coset C of a subgroup of ${\bf Z}^{d}$ a Cartesian coset if C equals the Cartesian product of d arithmetic progressions. Generalizing Mirsky–Newman, we show that a non-trivial disjoint family of Cartesian cosets with union ${\bf Z}^{d}$ always contains two cosets that differ only by translation. Where Mirsky–Newman’s proof (for d=1) uses complex analysis, we employ Fourier techniques. Relaxing the Cartesian requirement, for d>2 we provide examples where ${\bf Z}^{d}$ occurs as the disjoint union of four cosets of distinct subgroups (with one not Cartesian). Whether one can do the same for d=2 remains open.