Tiling with arbitrary tiles

Abstract

Let $T$ be a tile in ${\mathbb{Z}}^n$, meaning a finite subset of ${\mathbb{Z}}^n$. It may or may not tile ${\mathbb{Z}}^n$, in the sense of ${\mathbb{Z}}^n$ having a partition into copies of $T$. However, we prove that $T$ does tile ${\mathbb{Z}}^d$ for some $d$. This resolves a conjecture of Chalcraft.

23 pages, 19 figures; slightly updated