## Abstract

We obtain structural results on translational tilings of periodic functions in Z^{d} by finite tiles. In particular, we show that any level one tiling of a periodic set in Z^{2} must be weakly periodic (the disjoint union of sets that are individually periodic in one direction), but present a counterexample of a higher level tiling of Z^{2} that fails to be weakly periodic. We also establish a quantitative version of the two-dimensional periodic tiling conjecture which asserts that any finite tile in Z^{2} that admits a tiling, must admit a periodic tiling, by providing a polynomial bound on the period; this also gives an exponential-type bound on the computational complexity of the problem of deciding whether a given finite subset of Z^{2} tiles or not. As a byproduct of our structural theory, we also obtain an explicit formula for a universal period for all tilings of a one-dimensional tile.

Original language | English |
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Article number | 16 |

Journal | Discrete Analysis |

Volume | 2021 |

DOIs | |

State | Published - 2021 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2021. Rachel Greenfeld and Terence Tao

### Funding

*Partially supported by the Eric and Wendy Schmidt Postdoctoral Award. †Partially supported by NSF grant DMS-1764034 and by a Simons Investigator Award. 1All tilings in this paper are by translation; we do not consider tilings that involve rotation or reflection in addition to translation.

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

## Keywords

- Decidability
- Periodicity
- Tiling

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