Periods of sets of lengths: a quantitative result and an associated inverse problem

Wolfgang A. Schmid¹

¹ Institut für Mathematik und Wissenschaftliches Rechnen Karl-Franzens-Universität Heinrichstraße 36 8010 Graz, Austria

Colloquium Mathematicum, Tome 113 (2008) no. 1, pp. 33-53.

Résumé

The investigation of quantitative aspects of non-unique factorizations in the ring of integers of an algebraic number field gives rise to combinatorial problems in the class group of this number field. In this paper we investigate the combinatorial problems related to the function $\mathcal{P}(H,\mathcal{D}, M)(x)$, counting elements whose sets of lengths have period $\mathcal{D}$, for extreme choices of $\mathcal{D}$. If the class group meets certain conditions, we obtain the value of an exponent in the asymptotic formula of this function and results that imply oscillations of an error term.

Détail

DOI : 10.4064/cm113-1-4

Keywords: investigation quantitative aspects non unique factorizations ring integers algebraic number field gives rise combinatorial problems class group number field paper investigate combinatorial problems related function mathcal mathcal counting elements whose sets lengths have period mathcal extreme choices mathcal class group meets certain conditions obtain value exponent asymptotic formula function results imply oscillations error term