A counterexample to the periodic tiling conjecture

• 成果类型: Article
• 署名作者: Greenfeld, Rachel; Tao, Terence
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2024.200.1.5
• 发表日期: 2024
• 页码: 301-363
• 关键词: aperiodic set; tile; undecidability; nonperiodicity; subshifts; line

摘要

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.