Punctured intervals tile \mathbb Z³

2018/08/16 by Cambie, Stijn
#05B45 #05B50 #52C22 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.05433

Abstract

Extending the methods of Metrebian (2018), we prove that any symmetric punctured interval tiles \mathbb Z³. 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 ℤ^d. An affirmative answer would generalize a conjecture of Gruslys, Leader and Tan (2016).

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