Tiling Lattices with Sublattices, I

Abstract

Call a coset C of a subgroup of Zd 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 Zd 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 Zd 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. © 2010 Springer Science+Business Media, LLC.