The Buchweitz Set of a Numerical Semigroup

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URI: http://hdl.handle.net/10498/27845
DOI: 10.1007/s00574-022-00322-8
ISSN: 1678-7544
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2023-03Department
MatemáticasSource
Bulletin of the Brazilian Mathematical Society, Vol. 54, Núm. 1Abstract
Let A subset of 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+middotmiddotmiddot+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 subset of N is a Weierstrass semigroup, then B(N\S)= empty set . By constructing instances where this condition fails, Buchweitz disproved a longstanding conjecture by Hurwitz (Math Ann 41:403-442, 1893). In this paper, we prove that for any numerical semigroup S subset of N of genus g >= 2, the set B(N\S) is finite, of unbounded cardinality as S varies.
Subjects
Weierstrass numerical semigroup; Gapset; Additive combinatorics; Sumset growth; Freiman’s 3k; 3 theoremCollections
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