×

Periods of sets of lengths: a quantitative result and an associated inverse problem. (English) Zbl 1209.11096

It has been established by A. Geroldinger [Math. Z. 197, 505–529 (1988; Zbl 0618.12002)] that the set \(L(a)\) of lengths of factorizations of a non-unit \(a\) in the ring of integers \(R\) of an algebraic number field forms an almost arithmetical multiprogression (AAMP) [see e.g. Sect.4.2 in the book of A. Geroldinger and F. Halter-Koch, Non-unique factorizations. Algebraic, combinatorial and analytic theory. Boca Raton, FL: Chapman & Hall/CRC (2006; Zbl 1113.11002)]. There are three parameters associated with every AAMP, namely its difference \(d\), period \(D\) and bound \(M\). The author proves an asymptotic formula for the number of non-associated elements \(a\in R\) with \(|N(a)|\leq x\) for which the set \(L(a)\) is an AAMP with fixed parameters \(d\), \(D\) and \(M\).

MSC:

11R27 Units and factorization
11N64 Other results on the distribution of values or the characterization of arithmetic functions
20K01 Finite abelian groups
PDFBibTeX XMLCite
Full Text: DOI