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- W2018076385 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." @default.
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- W2018076385 date "2010-06-25" @default.
- W2018076385 modified "2023-09-26" @default.
- W2018076385 title "Tiling Lattices with Sublattices, I" @default.
- W2018076385 cites W2073667619 @default.
- W2018076385 doi "https://doi.org/10.1007/s00454-010-9272-1" @default.
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