Atomic density of arithmetical congruence monoids.

Consider the set M a , b = { n ∈ Z ≥ 1 : n ≡ a mod b } ∪ { 1 } for a , b ∈ Z ≥ 1 . If a 2 ≡ a mod b , then M a , b is closed under multiplication and known as an arithmetic congruence monoid (ACM). A non-unit n ∈ 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. [ABSTRACT FROM AUTHOR]