Abstract
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.
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Feldman, D., Propp, J. & Robins, S. Tiling Lattices with Sublattices, I. Discrete Comput Geom 46, 184–186 (2011). https://doi.org/10.1007/s00454-010-9272-1
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DOI: https://doi.org/10.1007/s00454-010-9272-1