The Buchweitz Set of a Numerical Semigroup

Abstract Let $$A \subset \mathbb Z$$ A ⊂ Z be a finite subset. We denote by $${{\,\mathrm{\mathcal {B}}\,}}(A)$$ B ( ) the set of all integers $$n \ge 2$$ n ≥ 2 such that $$|nA| > (2n-1)(|A|-1),$$ | > - 1 , where $$nA=A+\cdots +A$$ = + ⋯ denotes n -fold sumset A . The motivation to consider stems from Buchweitz’s discovery in 1980 if numerical semigroup $$S \subseteq N$$ S ⊆ N is Weierstrass semigroup, then {B}}\,}}(\mathbb N{\setminus } S) = \emptyset .$$ \ ∅ . By constructing instances this condition fails, Buchweitz disproved longstanding conjecture Hurwitz (Math Ann 41:403–442, 1893). In paper, we prove for any genus $$g 2,$$ g $$ finite, unbounded cardinality as S varies.