A counterexample to the periodic tiling conjecture (announcement)

TL;DR

This paper presents a counterexample disproving the periodic tiling conjecture in high dimensions, showing that some tiles can tile a lattice without doing so periodically, which challenges previous assumptions.

Contribution

The authors provide the first known counterexample to the periodic tiling conjecture for large dimensions, extending the disproof to Euclidean spaces and certain groups.

Findings

Counterexample exists in high-dimensional lattices

Disproves the conjecture for Euclidean spaces

Uses encoding of p-adically structured functions

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.