Tiling Lattices with Sublattices, II

Details Tiling Lattices with Sublattices, II Tiling Lattices with Sublattices, II

2010-06-02

arxiv.org/abs/1006.0472v2

Mathematics

Combinatorics

Scientific paper

Our earlier article proved 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. We re-prove this Theorem, this time using generating functions. We also show that for $d \geq 1$, not every finite tiling of $Z^d$ by lattices can be obtained from the trivial tiling by the process of repeatedly subdividing a tile into sub-tiles that are translates of one another.

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