On products of k atoms II

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.