## Abstract

The periodic tiling conjecture asserts that any finite subset of a lattice Z^{d} that tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large d, which also implies a disproof of the corresponding conjecture for Euclidean spaces R^{d}. In fact, we also obtain a counterexample in a group of the form Z^{2} x G_{0} for some finite abelian 2-group G_{0}. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "2-adically structured functions," in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist, but are all non-periodic.

Original language | English (US) |
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Pages (from-to) | 301-363 |

Number of pages | 63 |

Journal | Annals of Mathematics |

Volume | 200 |

Issue number | 1 |

DOIs | |

State | Published - 2024 |

### Funding

1.6. Acknowledgments. RG was partially supported by the AMIAS Membership and NSF grants DMS-2242871 and DMS-1926686. TT was partially supported by NSF grant DMS-1764034 and by a Simons Investigator Award. We thank Nishant Chandgotia, Asaf Katz, S\u00E9bastien Labb\u00E9and Misha Sodin for drawing our attention to some relevant references and to Emmanuel Jean-del for helpful comments. We are also grateful to the anonymous referee for many helpful suggestions that improved the exposition of this paper.

## Keywords

- periodicity
- tiling

## ASJC Scopus subject areas

- Mathematics (miscellaneous)