The Buchweitz set of a numerical semigroup

Abstract

Let $A\subset \mathbb{Z}$ be a finite subset. We denote by $\mathcal{B}(A)$ the set of all integers $n\ge 2$ such that $|nA|>(2n-1)(|A|-1)$, where $nA=A+\cdots +A$ denotes the $n$-fold sumset of $A$. The motivation to consider $\mathcal{B}(A)$ stems from Buchweitz's discovery in 1980 that if a numerical semigroup $S\subseteq \mathbb{N}$ is a Weierstrass semigroup, then $\mathcal{B}(\mathbb{N}\setminus S)=\varnothing$. By constructing instances where this condition fails, Buchweitz disproved a longstanding conjecture by Hurwitz (1893). In this paper, we prove that for any numerical semigroup $S\subset \mathbb{N}$ of genus $g\ge 2$, the set $\mathcal{B}(\mathbb{N}\setminus S)$ is finite, of unbounded cardinality as $S$ varies.

References in corpus (1)

• On reducible non-Weierstrass semigroups