Fibonacci-like growth of numerical semigroups of a given genus

Abstract

We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if $n_g$ is the number of numerical semigroups of genus $g$, we prove that $n_g$ tends to $S \phi^g$, where $\phi$ is the golden ratio, and $S$ is a constant, resolving several related conjectures concerning the growth of $n_g$. In addition, we show that the proportion of numerical semigroups of genus $g$ satisfying $f < 3m$ approaches 1 as $g \rightarrow \infty$, where $m$ is the multiplicity and $f$ is the Frobenius number.