On products of k atoms

Abstract.Let H be an atomic monoid. For $k \in {\Bbb N}$ let ${\cal V}_k (H)$ denote the set of all $m \in {\Bbb N}$ with the following property: There exist atoms (irreducible elements) u1, …, uk, v1, …, vm ∈ H with u1· … · uk = v1 · … · vm. We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets ${\cal V}_k (H)$ are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every $k \in {\Bbb N}$, max ${\cal V}_{2k+1} (H) = k \vert G\vert + 1$ which settles Problem 38 in [4].