Tiling Lattices with Sublattices, I

Abstract

We use Fourier methods to prove that if $n>1$ translates of sublattices of $Z^d$ tile $Z^d$, and all the sublattices are Cartesian products of arithmetic progressions, then two of the tiles must be translates of each other. This is a multi-dimensional generalization of the Mirsky-Newman Theorem.

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