The Golomb--Welch conjecture for lattice tilings

Abstract

We prove the Golomb--Welch conjecture for lattice tilings: for every $n\ge 3$ and $r\ge 2$, the integer lattice ${\mathbb{Z}}^n$ admits no lattice tiling by Lee balls $B_n(r)$ of radius $r$. Together with the classical constructions for $n\le 2$ 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 $2n\ge 6$ and diameter $r\ge 2$ has strictly fewer than $|B_n(r)|$ vertices.