Tiling Lattices with Sublattices, I

TL;DR

This paper proves that in multi-dimensional integer lattices, if the lattice is tiled by translates of certain structured sublattices, then at least two of these tiles are essentially the same, extending a classical theorem.

Contribution

It generalizes the Mirsky-Newman Theorem to higher dimensions for tiles formed by Cartesian products of arithmetic progressions.

Findings

If $n > 1$ translates of such sublattices tile $Z^d$, two are translates of each other.

The result applies to sublattices that are Cartesian products of arithmetic progressions.

The proof uses Fourier analysis methods.

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.