Density of Numerical sets associated to a Numerical semigroup

TL;DR

This paper investigates the density of numerical sets associated with numerical semigroups, identifying limits for the proportion of sets mapped to specific families as the Frobenius number grows large.

Contribution

It introduces a classification of numerical semigroup families and determines their asymptotic densities, extending understanding of the distribution of numerical sets.

Findings

The ratio of numerical sets mapped to a specific family converges to a positive limit.

The constants  sum to 1, accounting for almost all numerical sets asymptotically.

Identifies a collection of families with well-defined asymptotic densities.

Abstract

A numerical set is a co-finite subset of the natural numbers that contains zero. Its Frobenius number is the largest number in its complement. Each numerical set has an associated semigroup $A(T)=\{t\mid t+T\subseteq T\}$, which has the same Frobenius number as $T$. For a fixed Frobenius number $f$ there are $2^{f-1}$ numerical sets. It is known that there is a number $\gamma$ close to $0.484$ such that the ratio of these numerical sets that are mapped to $N_f=\{0\}\cup(f,\infty)$ is asymptotically $\gamma$. We identify a collection of families $N(D,f)$ of numerical semigroups such that for a fixed $D$ the ratio of the $2^{f-1}$ numerical sets that are mapped to $N(D,f)$ converges to a positive limit as $f$ goes to infinity. We denote the limit as $\gamma_D$, these constants sum up to $1$ meaning that they asymptotically account for almost all numerical sets.