Domination ratio of integer distance digraphs

An integer distance digraph is the Cayley graph $\mathrm{\Gamma }(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to some finite subset $S\subseteq \mathbb{Z}$. The domination ratio of $\mathrm{\Gamma }(\mathbb{Z},S)$ is the minimum density of a dominating set in $\mathrm{\Gamma }(\mathbb{Z},S)$. We establish some basic results on the domination ratio of $\mathrm{\Gamma }(\mathbb{Z},S)$ and precisely determine it when $S=\{s,t\}$ with $s$ dividing $t$.