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arXiv · 2610.33983

The Golomb--Welch conjecture for lattice tilings

Bocong Chen·Jing Huang·Gaojun Luo
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Abstract

We prove the Golomb--Welch conjecture for lattice tilings: for every $n\ge3$ and $r\ge2$, the integer lattice $\Z^n$ admits no lattice tiling by Lee balls $B_n(r)$ of radius $r$. Together with the classical constructions for $n\le2$ and for $r=1$, this determines all parameters for which a lattice Lee tiling exists. Equivalently, we show that every abelian Cayley graph of even degree $2n\ge6$ and diameter $r\ge2$ has strictly fewer than $|B_n(r)|$ vertices.

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Bocong Chen, Jing Huang, Gaojun Luo. 2026-10-25. The Golomb--Welch conjecture for lattice tilings. https://arxiv.org/abs/2610.33983

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