A counterexample to the periodic tiling conjecture

Rachel Greenfeld, Terence Tao

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

The periodic tiling conjecture asserts that any finite subset of a lattice Zd 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 Rd. In fact, we also obtain a counterexample in a group of the form Z2 x G0 for some finite abelian 2-group G0. 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 languageEnglish (US)
Pages (from-to)301-363
Number of pages63
JournalAnnals of Mathematics
Volume200
Issue number1
DOIs
StatePublished - 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)

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