Periodicity and decidability of translational tilings by rational polygonal sets

Jaume de Dios Pont, Jan Grebík, Rachel Greenfeld*, José Madrid

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The periodic tiling conjecture asserts that if a region Σ⊂Rd tiles Rd by translations then it admits at least one fully periodic tiling. This conjecture is known to hold in R, and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic tiling conjecture for polygonal sets: bounded open sets in R2 whose boundary is a finite union of line segments. We prove the periodic tiling conjecture for any polygonal tile whose vertices are rational. As a corollary of our argument, we also obtain the decidability of tilings by rational polygonal sets. Moreover, we prove that any translational tiling by a rational polygonal tile is weakly-periodic, i.e., can be partitioned into finitely many singly-periodic pieces.

Original languageEnglish (US)
Article number125620
JournalExpositiones Mathematicae
DOIs
StateAccepted/In press - 2024

Funding

JD was partially supported by a UCLA Dissertation Year Fellowship award and the Simons Collaborations in MPS grant 563916 . JG was supported by MSCA Postdoctoral Fellowships 2022 HORIZON-MSCA-2022-PF-01-01 project BORCA grant agreement number 101105722 . RG was supported by the Association of Members of the Institute for Advanced Study (AMIAS) and by NSF grant DMS-2242871 . JM was supported by the AMS Stefan Bergman Fellowship . We are grateful to Terence Tao and the anonymous referee for their helpful suggestions, which improved the exposition of this paper. JD was partially supported by a UCLA Dissertation Year Fellowship award, United States of America and the Simons Collaborations in MPS, United States of America grant 563916. JG was supported by MSCA Postdoctoral Fellowships 2022 HORIZON-MSCA-2022-PF-01-01 project BORCA grant agreement number 101105722. RG was supported by the Association of Members of the Institute for Advanced Study (AMIAS), United States of America and by National Science Foundation, United States of America grant DMS-2242871. JM was supported by the AMS Stefan Bergman Fellowship, United States of America.

Keywords

  • Decidability
  • Periodicity
  • Tiling

ASJC Scopus subject areas

  • General Mathematics

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