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 $\{ -k,\ldots,k\} \subset \mathbb{ Z}$ 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 $\mathbb{ Z}^4$.
Categories: math.CO