On the asymptotic behavior of unions of sets of lengths in atomic monoids

Abstract

Let $M$ be a commutative cancellative atomic monoid. We use unions of sets of lengths in $M$ to construct the $\mathcal{V}$-Delta set of $M$. We first derive some basic properties of $\mathcal{V}$-Delta sets and then show how they offer a method to investigate the asymptotic behavior of the sizes of unions of sets of lengths.