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Journal Articles Boletim da Sociedade Brasileira de Matemática / Bulletin of the Brazilian Mathematical Society Year : 2023

The Buchweitz set of a numerical semigroup

Abstract

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.
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Dates and versions

hal-03032801 , version 1 (01-12-2020)

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Shalom Eliahou, Juan Ignacio García-García, Daniel Marín-Aragón, Alberto Vigneron-Tenorio. The Buchweitz set of a numerical semigroup. Boletim da Sociedade Brasileira de Matemática / Bulletin of the Brazilian Mathematical Society, 2023, 54 (4), ⟨10.1007/s00574-022-00322-8⟩. ⟨hal-03032801⟩
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