An asymptotic result concerning a question of Wilf

TL;DR

This paper proves that for fixed embedding dimension, the ratio of certain semigroup elements approaches a bound related to Wilf's conjecture, holding for all but finitely many cases as the semigroup size grows.

Contribution

It establishes an asymptotic version of Wilf's conjecture, showing the ratio bound holds for all but finitely many numerical semigroups with fixed embedding dimension.

Findings

The ratio c'/c approaches at least 1/e as semigroup size increases.

The result holds for all but finitely many semigroups with fixed embedding dimension.

Provides an asymptotic validation of Wilf's conjecture.

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 $e(\Lambda) = k$.