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On invariants related to non-unique factorizations in block monoids and rings of algebraic integers. (English) Zbl 1108.11074
Summary: Let \(K\) be a number field, \(R\) its ring of integers and \(H\) the set of non-zero principal ideals of \(R\). For each positive integer \(k\) the set \(\mathcal {B}_k(H)\subset H\) denotes the set of principal ideals for which the associated block has at most \(k\) different factorizations. For the counting functions associated to these sets asymptotic formulae are known. These formulae involve constants that just depend on the ideal class group \(G\) of \(R\). Starting from a known combinatorial description for these constants, we use tools from additive group theory, in particular the notion of Davenport’s constant and a classical addition theorem, to investigate them. We determine their precise value in case \(G\) is an elementary group or a cyclic group of prime power order. For arbitrary \(G\) we derive (explicit) lower bounds.

MSC:
11N64 Other results on the distribution of values or the characterization of arithmetic functions
11R27 Units and factorization
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20K01 Finite abelian groups
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