The periodic tiling conjecture for translational tiles

Let $\Omega\subset\mathbb R^d$ be a set of finite, positive measure. A translational tiling of $\mathbb R^d$ is a collection of translates of $\Omega$ covering almost every point exactly once; it is periodic when its translation set is invariant under a full-rank lattice. The periodic tiling conjecture. If $\Omega$ tiles $\mathbb R^d$ by translations, then it admits at least one periodic tiling. The conjecture is a central question in translational tiling theory, but the source notes that it was disproved in high dimensions; the connected-set case was the motivation for the paper's theorem.