A realization result for systems of sets of lengths

Abstract

Let ${\mathcal{L}}^{\ast }$ be a family of finite subsets of ${\mathbb{N}}_0$ having the following properties. (a). $\{0\},\{1\}\in {\mathcal{L}}^{\ast }$ and all other sets of ${\mathcal{L}}^{\ast }$ lie in ${\mathbb{N}}_{\ge 2}$. (b). If $L_1,L_2\in {\mathcal{L}}^{\ast }$, then the sumset $L_1+L_2\in {\mathcal{L}}^{\ast }$. We show that there is a Dedekind domain $D$ whose system of sets of lengths equals ${\mathcal{L}}^{\ast }$.

To appear in Israel Journal of Mathematics