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 language | English |
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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 |
Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2024 Department of Mathematics, Princeton University
Keywords
- periodicity
- tiling