Atomic density of arithmetical congruence monoids

Consider the set $M_{a,b}=\{n\in {\mathbb{Z}}_{\ge 1}:n\equiv ab\}\cup \{1\}$ for $a,b\in {\mathbb{Z}}_{\ge 1}$. If $a^2\equiv ab$, then $M_{a,b}$ is closed under multiplication and known as an arithmetic congruence monoid (ACM). A non-unit $n\in M_{a,b}$ is an atom if it cannot be expressed as a product of non-units, and the atomic density of $M_{a,b}$ is the limiting proportion of elements that are atoms. In this paper, we characterize the atomic density of $M_{a,b}$ in terms of $a$ and $b$.