Note on a question of Wilf

TL;DR

This paper investigates Wilf's conjecture on numerical semigroups, providing the first explicit lower bound for the ratio rac{f+1-g}{f+1} in terms of the embedding dimension for e ≥ 4.

Contribution

It derives a new explicit lower bound for the Wilf ratio based on results from Zhai, extending the understanding for cases where e ≥ 4.

Findings

Established a lower bound for rac{f+1-g}{f+1} in terms of e

First explicit bound connecting Wilf's ratio and embedding dimension

Provides a quantitative measure for Wilf's conjecture in new cases

Abstract

Let $S$ 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 $e=3$. For $e\geq4$, we derive from results of Zhai [5] the following (substantially weaker) lower bound \[\frac{f+1-g}{f+1}>\left(\frac{2N+1}{(2N+2)(e-2)}\right)^e\text{ with }\lfloor N\rfloor=104978\,.\] To the best of our knowledge this is the first explicit lower bound for $\frac{f+1-g}{f+1}$ in terms of the embedding dimension.