A counterexample to the periodic tiling conjecture

By Rachel Greenfeld, Terence Tao. Annals of Mathematics, 2022

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The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which 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 $\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group $G_0$. 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.

Subfields: Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/567e336347faa5b2d927b123f0c1820f56368625