The Structure of Translational Tilings in Zd

Rachel Greenfeld, Terence Tao

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

We obtain structural results on translational tilings of periodic functions in Zd by finite tiles. In particular, we show that any level one tiling of a periodic set in Z2 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 Z2 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 Z2 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 Z2 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 languageEnglish
Article number16
JournalDiscrete Analysis
Volume2021
DOIs
StatePublished - 2021
Externally publishedYes

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.

FundersFunder number
National Science FoundationDMS-1764034

    Keywords

    • Decidability
    • Periodicity
    • Tiling

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