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When $\mathbb{Z}^d$ is represented as a finite disjoint union of translatedinteger sublattices, the translated sublattices must possess some specialproperties. Such a representation is called a \emph{lattice tiling}. We developa theoretical framework, based on multiple residues and dual groups, to providea set of necessary and sufficient conditions for such a lattice tiling toexist. We also investigate the question of when a lattice tiling must possessat least two translated sublattices which are translates of one another.

Tiling the integer lattice with translated sublattices