Atomic density of arithmetical congruence monoids

Consider the set \(M_{a,b} = \{n \in \mathbb {Z}_{\ge 1}: n \equiv a \bmod b\} \cup \{1\}\) for \(a, b \in \mathbb {Z}_{\ge 1}\). If \(a^2 \equiv a \bmod b\), 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.