The Lagarias--Wang periodic tiling conjecture

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 1

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).

Sources & referencesView supporting material

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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Equivalent formulations 1

Lagarias–Wang periodic tiling conjecture

Primary source

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