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

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 \[\frac{f+1-g}{f+1}\geq\frac2{e^2-e+2}\text{ for }e\geq4\,.\]