Tiling with punctured intervals

TL;DR

This paper proves that punctured intervals, a specific type of finite subset of integers, can tile higher-dimensional integer lattices, specifically showing they tile Z^4, answering a previously open question.

Contribution

It demonstrates that punctured intervals tile Z^4, reducing the previously known dimension requirement that grows with the size of the interval.

Findings

Punctured intervals tile Z^4.

The dimension needed for tiling does not tend to infinity with the interval size.

Answers an open question about tiling dimensions for 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 $\{-k,\ldots,k\} \subset \mathbb{Z}$ 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 $\mathbb{Z}^4$.