Tiling with punctured intervals

Abstract

It was shown by Gruslys, Leader and Tan that any finite subset of ${\mathbb{Z}}^n$ tiles ${\mathbb{Z}}^d$ for some $d$. The first non-trivial case is the punctured interval, which consists of the interval $\{-k,\dots ,k\}\subset \mathbb{Z}$ with its middle point removed: they showed that this tiles ${\mathbb{Z}}^d$ for $d=2k^2$, and they asked if the dimension needed tends to infinity with $k$. In this note we answer this question: we show that, perhaps surprisingly, every punctured interval tiles ${\mathbb{Z}}^4$.

7 pages