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).

arXiv comment8 pages, 7 figures. About the update: the main result is now proven more general, explaining the change in the title. Some further details on this change are added in a note

References in corpus (1)

• Tiling with punctured intervals