s proved by Gruslys, Leader and Tan [@gltan16]. The first non-trivial case is the *punctured interval* $T = \underbrace{\texttt{XXXXX}}_{k}\!\texttt{.}\!\underbrace{\texttt{XXXXX}}_{k}$. The authors of [@gltan16] showed that $T$ tiles $\mathbb{Z}^d$ for $d = 2k^2$, but they were unable to prove that the smallest required dimension $d$ was quadratic in $k$, or even that $d \to \infty$ as $k \to \infty$. They therefore asked the following question:

Let $T$ be the punctured interval $\underbrace{\texttt{\emph{XXXXX}}}_{k}\!\texttt{.}\!\underbrace{\texttt{\emph{XXXXX}}}_{k}$, and let $d$ be the least number such that $T$ tiles $\mathbb{Z}^d$. Does $d \to \infty$ as $k \to \infty$?

In this paper we will show that, rather unexpectedly, $d$ does not tend to $\infty$:

\[mainthm\] Let $T$ be the punctured interval $\underbrace{\texttt{\emph{XXXXX}}}_{k}\!\texttt{.}\!\underbrace{\texttt{\emph{XXXXX}}}_{k}$. Then $T$ tiles $\mathbb{Z}^4$. Furthermore, if $k$ is odd or congruent to $4 \pmod 8$, then $T$ tiles $\mathbb{Z}^3$.

We have already noted that `X.X` tiles $\mathbb{Z}$, and `XX.XX` tiles $\mathbb{Z}^2$ but not $\mathbb{Z}$. It can be shown via case analysis that, for $k \geq 3$, the tile $T$ does not tile $\mathbb{Z}^2$. However, this proof is tedious and provides little insight, and since it is not the focus of this paper, we omit it. For odd $k \geq 3$ and for $k \equiv 4 \pmod 8$, 3 is therefore the least $d$ such that $T$ tiles $\mathbb{Z}^d$. For the remaining cases, namely $k \equiv 0, 2, 6 \pmod 8$, $k \geq 6$, it is unknown whether the least possible $d$ is 3 or 4.

In this paper, we will first prove the result for odd $k$. This will introduce some key ideas, which we will develop to prove the result for general $k$, and then to improve the dimension from 4 to 3 for $k \equiv 4 \pmod 8$.

Finally, we give some background. Tilings of $\mathbb{Z}^2$ by polyominoes (edge-connected tiles in $\mathbb{Z}^2$) have been thoroughly investigated. For example, Golomb [@golomb70] showed that results of Berger [@berger