An asymptotic result concerning a question of Wilf

Abstract

Let $\Lambda$ be a numerical semigroup with embedding dimension $e(\Lambda)$. Define $c(\Lambda)$ to be one plus the largest integer not in $\Lambda$, and define $c'(\Lambda)$ to be the number of elements in $\Lambda$ less than $c(\Lambda)$. It was asked by Wilf whether $\frac{c'(\Lambda)}{c(\Lambda)} \ge \frac{1}{e(\Lambda)}$ always holds. We prove an asymptotic version of this conjecture: we show that for a fixed positive integer $k$ and any $\epsilon > 0$, the inequality $\frac{c'(\Lambda)}{c(\Lambda)} \ge \frac{1}{k} - \epsilon$ holds for all but finitely many numerical semigroups $\Lambda$ satisfying $e(\Lambda) = k$.