Factorization length distribution for affine semigroups I: numerical semigroups with three generators

TL;DR

This paper investigates the distribution of factorization lengths in numerical semigroups with three generators, revealing that the median length asymptotically approaches an often irrational number, using analytical and probabilistic methods.

Contribution

It introduces a novel analysis of intermediate factorization invariants, such as median length, in affine semigroups, expanding understanding beyond extremal invariants.

Findings

Asymptotic median factorization length is often irrational.

Uses analysis and probability techniques to describe invariant behavior.

Provides new insights into the distribution of factorization lengths.

Abstract

Most factorization invariants in the literature extract extremal factorization behavior, such as the maximum and minimum factorization lengths. Invariants of intermediate size, such as the mean, median, and mode factorization lengths are more subtle. We use techniques from analysis and probability to describe the asymptotic behavior of these invariants. Surprisingly, the asymptotic median factorization length is described by a number that is usually irrational.