On products of k atoms

Let H be an atomic monoid. For k ? Bbb Nk in {Bbb N} let V_k (H){cal V}_k (H) denote the set of all m ? Bbb Nm in {Bbb N} with the following property: There exist atoms (irreducible elements) u ₁, …, u _k , v ₁, …, v _m ∈ H with u ₁· … · u _k = v ₁ · … · v _m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets V_k (H){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 ? Bbb Nk in {Bbb N}, max V_2k+1 (H) = k |G|+ 1{cal V}_{2k+1} (H) = k vert Gvert + 1 which settles Problem 38 in [4].