On products of k atoms II (1503.06164v1)

Abstract: Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor (for example, rings of integers in algebraic number fields or holomorphy rings in algebraic function fields). For $k \in \mathbb N$, let $\mathcal U_k (H)$ denote the set of all $m \in \mathbb N$ with the following property: There exist atoms $u_1, ..., u_k, v_1, ..., v_m \in H$ such that $u_1 \cdot ... \cdot u_k = v_1 \cdot ...\cdot v_m$. Furthermore, let $\lambda_k (H) = \min \mathcal U_k (H)$ and $\rho_k (H) = \sup \mathcal U_k (H)$. The sets $\mathcal U_k (H) \subset \mathbb N$ are intervals which are finite if and only if $G$ is finite. Their minima $\lambda_k (H)$ can be expressed in terms of $\rho_k (H)$. The invariants $\rho_k (H)$ depend only on the class group $G$, and in the present paper they are studied with new methods from Additive Combinatorics.