A counterexample to the periodic tiling conjecture (announcement)

Abstract

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. We announce here a disproof of 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 $G_0$. Our methods rely on encoding a certain class of "$p$-adically structured functions" in terms of certain functional equations.