A lower bound for the Wilf density, deduced from a result of Zhai

TL;DR

This paper establishes a new lower bound for the Wilf density of numerical semigroups with embedding dimension at least four, improving upon previous bounds and based on Zhai's results.

Contribution

It derives a novel lower bound for Wilf density for numerical semigroups with embedding dimension e ≥ 4, extending known results for e=2 and e=3.

Findings

For e=2 and e=3, the Wilf conjecture holds.

A new lower bound for e ≥ 4: (f+1-g)/(f+1) ≥ 2/(e^2 - e + 2).

The bound improves understanding of the Wilf density's behavior.

Abstract

Let $S\neq\mathbb N$ be a numerical semigroup with Frobenius number $f$, genus $g$ and embedding dimension $e$. In 1978 Wilf asked the question, whether $\frac{f+1-g}{f+1}\geq\frac1e$. As is well known, this holds in the cases $e=2$ and $e=3$. From Zhai's results in [5] we derive \[\frac{f+1-g}{f+1}\geq\frac2{e^2-e+2}\text{ for }e\geq4\,.\]