Periods of sets of lengths: a quantitative result and an associated inverse problem
Volume 113 / 2008
Colloquium Mathematicum 113 (2008), 33-53
MSC: Primary 11R27; Secondary 11N64, 20K01.
DOI: 10.4064/cm113-1-4
Abstract
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.
