The Lagarias--Wang periodic tiling conjecture

About 16 years old · traced to 2010

Let $V$ be a finite tile in $\mathbb{Z}^n$ . A tiling by translations is fully periodic if its translation set has a finite-index period subgroup. Lagarias--Wang conjecture. If $V$ tiles $\mathbb{Z}^n$ by translations, then $V$ admits a fully periodic tiling, equivalently a $q$ -periodic tiling for sufficiently large $q$ . The paper explains that this conjecture would make the weak and strong Golomb--Welch conjectures equivalent, but gives no resolution of it.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Lagarias–Wang periodic tiling conjecture
Let $d>1$ and let $cOmegacsubset cmathbb{Z}^d$ be a finite set. Lagarias–Wang's conjecture. If $cOmega$ tiles $cmathbb{Z}^d$ by translation, then it admits a periodic tiling. Here a periodic tiling means that the tiling set is invariant under a finite-index subgroup of $cmathbb{Z}^d$ . The conjecture is known in dimension one, but the supplied context says that Greenfeld and Tao disproved the periodic tiling conjecture in sufficiently large dimensions.
source: Shilei Fan and Tao Zhang, “Periodicity of tiles in finite Abelian groups”, arXiv:2408.12901 (2025).

References

Primary source

Peter Horak and Dongryul Kim, “50 Years of the Golomb–Welch Conjecture”, arXiv:1706.03589 (2018).

Additional references

2 papers in this index state this conjecture (2010–2017). The statement above is taken from the most recent of them; the others are arXiv:1009.3799.

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Progress summary

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RemarkAI-assistedClaimed by OpenAI. The manuscript claims a finite translational tile in Z^3 that tiles Z^3 but admits no fully periodic tiling, meaning none has a finite-index period subgroup. This gives a claimed three-dimensional counterexample to the page’s Lagarias–Wang formulation. It also claims the same absence of full periodicity for the unit-cube thickening in R^3 with arbitrary real translations; exclusion of every single nonzero period is not asserted.See full solutionHide full solution

Claimed by OpenAI. The manuscript claims a finite translational tile in Z^3 that tiles Z^3 but admits no fully periodic tiling, meaning none has a finite-index period subgroup. This gives a claimed three-dimensional counterexample to the page’s Lagarias–Wang formulation. It also claims the same absence of full periodicity for the unit-cube thickening in R^3 with arbitrary real translations; exclusion of every single nonzero period is not asserted.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf

OpenAI-155-01-A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three.pdf465,691 bytesOpen

Lagarias–Wang periodic tiling conjecture

Primary source

Additional references

Progress summary