The Buchweitz set of a numerical semigroup
Résumé
Let A ⊂ Z be a finite subset. We denote by B(A) the set of all integers n ≥ 2 such that |nA| > (2n − 1)(|A| − 1), where nA = A + • • • + A denotes the n-fold sumset of A. The motivation to consider B(A) stems from Buchweitz's discovery in 1980 that if a numerical semigroup S ⊆ N is a Weierstrass semigroup, then B(N \ S) = / 0. 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 ⊂ N of genus g ≥ 2, the set B(N \ S) is finite, of unbounded cardinality as S varies.
Origine | Fichiers produits par l'(les) auteur(s) |
---|