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Erschienen in:
16.06.2018 | Original Paper
Three questions of Bertram on locally maximal sum-free sets
Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 2/2019
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Abstract
Let G be a finite group, and S a sum-free subset of G. The set S is locally maximal in G if S is not properly contained in any other sum-free set in G. If S is a locally maximal sum-free set in a finite abelian group G, then \(G=S\cup SS\cup SS^{-1}\cup \sqrt{S}\), where \(SS=\{xy|~x,y\in S\}\), \(SS^{-1}=\{xy^{-1}|~x,y\in S\}\) and \(\sqrt{S}=\{x\in G|~x^2\in S\}\). Each set S in a finite group of odd order satisfies \(|\sqrt{S}|=|S|\). No such result is known for finite abelian groups of even order in general. In view to understanding locally maximal sum-free sets, Bertram asked the following questions: In this paper, we answer question (i) in the negative, then (ii) and (iii) in affirmative.
(i)
Does S locally maximal sum-free in a finite abelian group imply \(|\sqrt{S}|\le 2|S|\)?
(ii)
Does there exist a sequence of finite abelian groups G and locally maximal sum-free sets \(S\subset G\) such that \(\frac{|SS|}{|S|}\rightarrow \infty \) as \(|G|\rightarrow \infty \)?
(iii)
Does there exist a sequence of abelian groups G and locally maximal sum-free sets \(S\subset G\) such that \(|S|<c|G|^{\frac{1}{2}}\) as \(|G|\rightarrow \infty \), where c is a constant?