A counterexample to the periodic tiling conjecture

Annals of Mathematics · 2024 · 16 citations

• Mathematics
• Combinatorics
• Discrete mathematics

The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ 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 $\mathbb{R}^d$. 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.

Elias M. Stein (1931–2018)

Notices of the American Mathematical Society · 2021

• Philosophy
• Mathematical economics
• Mathematics

2018) had a profound influence on the field of analysis. He developed tools that are now indispensable, expanded and clarified major theories, and introduced new classes of questions that continue to stimulate research today. In addition, his singular skills as a mentor and expositor left a legacy of dozens of PhD students, hundreds of mathematical descendants, and thousands of loyal readers.