On the asymptotic behavior of unions of sets of lengths in atomic monoids. (Q940964)

Let \(M\) be a commutative multiplicative cancellative atomic monoid. For \(a\in M^*\) let \(L(a)\subset\mathbb{N}\) denote the set of all lengths of factorizations of \(a\) into atoms. For \(n\in\mathbb{N}\), let \(\mathcal V(n)\) be the union of all sets \(L(a)\) for \(a\in M^*\) with \(n\in L(a)\) and \(\Phi(n)=|\mathcal V(n)|\). Let \(\rho(M)\) denote the elasticity of \(M\) (for basic concepts and results of factorization theory see [\textit{A. Geroldinger} and \textit{F. Halter-Koch}, Non-unique factorizations. Algebraic, combinatorial and analytic theory. Pure Appl. Math. 278. Boca Raton: Chapman \& Hall/CRC (2006; Zbl 1113.11002)]). If \(\rho(M)<\infty\), then \(\Phi(n)<\infty\) for all \(n\in\mathbb{N}\), and the authors provide upper and lower bounds for the \(\liminf\) and the \(\limsup\) of \(\Phi(n)/n\) in terms of the maximal and minimal size of gaps in some set \(\mathcal V(n)\). If \(M\) is a Krull monoid with finite class group \(G\) such that every class contains a prime, there is a more precise result concerning the limit of \(\Phi(n)/n\) in terms of the Davenport constant of \(G\) [\textit{S. T. Chapman} and \textit{W. W. Smith}, J. Number Theory 43, No. 1, 24-30 (1993; Zbl 0765.11043)].