Tiling Lattices with Sublattices, II

TL;DR

This paper re-proves a theorem about tiling lattices with sublattices using generating functions and explores the limitations of constructing finite lattice tilings through subdivision processes.

Contribution

It provides a new proof of an existing theorem using generating functions and analyzes the constraints of building tilings via subdivision.

Findings

Re-proved the theorem using generating functions.

Not all finite lattice tilings can be generated by subdivision.

Identified limitations in tiling constructions.

Abstract

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.