Abstract
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.
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Acknowledgements
The authors would like to thank Vadim Ponomarenko for several helpful conversations and Paul Pollack for directing us to an appropriate citation for Lemma 2.1. We would also like to thank Adam Hoyt, Yuze Luan, and Melanie Zhang for their early computational work on the project.
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Communicated by Nathan Kaplan.
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Olsson, N., O’Neill, C. & Rawling, D. Atomic density of arithmetical congruence monoids. Semigroup Forum 108, 432–437 (2024). https://doi.org/10.1007/s00233-024-10426-w
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DOI: https://doi.org/10.1007/s00233-024-10426-w