Atomic density of arithmetical congruence monoids

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