## Abstract

We construct an example of a group G= Z^{2}× G for a finite abelian group G, a subset E of G, and two finite subsets F_{1}, F_{2} of G, such that it is undecidable in ZFC whether Z^{2}× E can be tiled by translations of F_{1}, F_{2}. In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of E by the tiles F_{1}, F_{2}, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in Z^{2}). A similar construction also applies for G= Z^{d} for sufficiently large d. If one allows the group G to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.

Original language | English |
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Pages (from-to) | 1652-1706 |

Number of pages | 55 |

Journal | Discrete and Computational Geometry |

Volume | 70 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2023 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2023, The Author(s).

### Funding

RG was partially supported by the Eric and Wendy Schmidt Postdoctoral Award. TT was partially supported by NSF grant DMS-1764034 and by a Simons Investigator Award. We gratefully acknowledge the hospitality and support of the Hausdorff Institute for Mathematics where a significant portion of this research was conducted. We thank David Roberts for drawing our attention to the reference [32], Hunter Spink for drawing our attention to the reference [15], Jarkko Kari for drawing our attention to the references [18 , 20 , 24], and Zachary Hunter for further corrections. We are also grateful to the anonymous referee for several suggestions that improved the exposition of this paper. RG was partially supported by the Eric and Wendy Schmidt Postdoctoral Award. TT was partially supported by NSF grant DMS-1764034 and by a Simons Investigator Award. We gratefully acknowledge the hospitality and support of the Hausdorff Institute for Mathematics where a significant portion of this research was conducted. We thank David Roberts for drawing our attention to the reference [], Hunter Spink for drawing our attention to the reference [], Jarkko Kari for drawing our attention to the references [, , ], and Zachary Hunter for further corrections. We are also grateful to the anonymous referee for several suggestions that improved the exposition of this paper.

Funders | Funder number |
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National Science Foundation | DMS-1764034 |

Hausdorff Research Institute for Mathematics |

## Keywords

- Aperiodic tiling
- Decidability
- Translational tiling