Some asymptotic results on $p$-lengths of factorizations for numerical semigroups and arithmetical congruence monoids

A factorization of an element $x$ in a monoid $(M, \cdot)$ is an expression of the form $x = u_1^{z_1} \cdots u_k^{z_k}$ for irreducible elements $u_1, \ldots, u_k \in M$, and the length of such a factorization is $z_1 + \cdots + z_k$. We introduce the notion of $p$-length, a generalized notion of factorization length obtained from the $\ell_p$-norm of the sequence $(z_1, \ldots, z_k)$, and present asymptotic results on extremal $p$-lengths of factorizations for large elements of numerical semigroups (additive submonoids of $\mathbb Z_{\ge 0}$) and arithmetical congruence monoids (certain multiplicative submonoids of $\mathbb Z_{\ge 1}$). Our results, inspired by analogous results for classical factorization length, demonstrate the types of combinatorial statements one may hope to obtain for sufficiently nice monoids, as well as the subtlety such asymptotic questions can have for general monoids.