  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.
