An asymptotic result concerning a question of Wilf

Abstract

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

9 pages, submitted to Semigroup Forum