Let Omega⊂ R^d be a set of finite, positive measure. A translational tiling of 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 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.

The periodic tiling conjecture for translational tiles

The conjecture asked whether every translational tile has a periodic tiling. Greenfeld and Tao announced a counterexample in 2022, first for discrete groups and then for measurable subsets of $\mathbb{R}^d$ .

Known results

Bhattacharya (2020) proved the discrete two-dimensional case: every finite tile in $\mathbb{Z}^2$ admitting a tiling admits a periodic tiling.
The conjecture holds in $\mathbb{R}$ .
It holds for convex domains in every dimension.
It holds for topological disks and rational polygonal tiles in $\mathbb{R}^2$ .

High-dimensional counterexamples, 2022–2024

Greenfeld and Tao constructed finite tiles in $\mathbb{Z}^d$ and measurable tiles in $\mathbb{R}^d$ whose tilings are all non-periodic, for sufficiently large $d$ ; the result appeared in the 2024 volume of the Annals of Mathematics. A 2023 follow-up claims connected counterexamples, in sufficiently large dimension, by converting a disconnected example into one in dimension $d+2$ .

Current status (as of September 2026): The general conjecture is claimed false in sufficiently high dimensions, including for connected tiles, while the low-dimensional cases beyond the established classes remain open.

The periodic tiling conjecture for translational tiles

Primary source

Progress summary

Solutions 0