2020/11/18 by S. Eliahou, Shalom Eliahou, J. I. García-García +6
Mathematics · #11P70 #14H55 #20M14 #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.AC #math.CO #math.NT #msc:11P70 #msc:14H55 #msc:20M14
paper · pdf · doi:10.48550/arxiv.2011.09187
openalex publication_date 2020/11/18 · arxiv created 2020/11/24 · arxiv updated 2020/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A ⊂ \mathbb 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 ⊆ \mathbb N is a Weierstrass semigroup, then B(\mathbb N ∖ S) = ∅. 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 ⊂ \mathbb N of genus g ≥ 2, the set B(\mathbb N ∖ S) is finite, of unbounded cardinality as S varies.