Punctured intervals tile $\mathbb Z^3$

Abstract

Extending the methods of Metrebian (2018), we prove that any symmetric punctured interval tiles $\mathbb Z^3$. This solves two questions of Metrebian and completely resolves a question of Gruslys, Leader and Tan. We also pose a question that asks whether there is a relation between the genus $g$ (number of holes) in a one-dimensional tile $T$ and a uniform bound $d$ such that $T$ tiles $\mathbb{Z}^d$. An affirmative answer would generalize a conjecture of Gruslys, Leader and Tan (2016).