Terence Tao@QuantaMagazine just published their article on the #PeriodicTilingConjecture, which was recently disproven by Rachel Greenfeld and myself. https://www.quantamagazine.org/nasty-geometry-breaks-decades-old-tiling-conjecture-20221215/
Rachel Greenfeld and I have just uploaded our paper on a counterexample to the #PeriodicTilingConjecture to the arXiv at https://arxiv.org/abs/2211.15847 . I also have a blog post on this paper at https://terrytao.wordpress.com/2022/11/29/a-counterexample-to-the-periodic-tiling-conjecture-2/
A counterexample to the periodic tiling conjecture
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.