Tiling with arbitrary tiles

Vytautas Gruslys; Imre Leader; Ta Sheng Tan

Tiling with arbitrary tiles

Combinatorics 2016-08-23 v2

Authors: Vytautas Gruslys , Imre Leader , Ta Sheng Tan

Abstract

Let $T$ be a tile in $\mathbb{Z}^n$ , meaning a finite subset of $\mathbb{Z}^n$ . It may or may not tile $\mathbb{Z}^n$ , in the sense of $\mathbb{Z}^n$ having a partition into copies of $T$ . However, we prove that $T$ does tile $\mathbb{Z}^d$ for some $d$ . This resolves a conjecture of Chalcraft.

Keywords

tilings

Cite

@article{arxiv.1505.03697,
  title  = {Tiling with arbitrary tiles},
  author = {Vytautas Gruslys and Imre Leader and Ta Sheng Tan},
  journal= {arXiv preprint arXiv:1505.03697},
  year   = {2016}
}

Comments

23 pages, 19 figures; slightly updated

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