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 2 N, let Uk(H) denote the set of all m 2 N with the following property: There exist atoms u1, …, uk, v1, …, vm 2 H such that u1 ¡ … ¡ uk = v1 ¡ … ¡ vm. Furthermore, let λk (H) = min Uk(H) and ρk (H) = sup Uk(H). The sets Uk(H) ⊂ N are intervals which are finite if and only if G is finite. Their minima λk (H) can be expressed in terms of ρk (H). The invariants ρk (H) depend only on the class group G, and in the present paper they are studied with new methods from Additive Combinatorics.

Publication Title

Moscow Journal of Combinatorics and Number Theory

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