A realization theorem for sets of distances

Let $H$ be an atomic monoid. The set of distances $\mathrm{\Delta }(H)$ of $H$ is the set of all $d\in \mathbb{N}$ with the following property: there are irreducible elements $u\mathrm{\_}1,\dots ,u\mathrm{\_}k,v\mathrm{\_}1\dots ,v\mathrm{\_}k+d$ such that $u\mathrm{\_}1\cdot \dots \cdot u\mathrm{\_}k=v\mathrm{\_}1\cdot \dots \cdot v\mathrm{\_}k+d$ but $u_1 \cdot \ldots \cdot u_k$cannotbewrittenasaproductof$\ell$ irreducible elements for any $\mathrm{\ell }\in \mathbb{N}$ with $k<\mathrm{\ell }<k+d$. It is well-known (and easy to show) that, if $\mathrm{\Delta }(H)$ is nonempty, then $\min \Delta (H) = \gcd \Delta (H)$. In this paper we show conversely that for every finite nonempty set $\mathrm{\Delta }\subset \mathbb{N}$ with $\min \Delta = \gcd \Delta$thereisafinitelygeneratedKrullmonoid$H$suchthat$\Delta (H)=\Delta$.