Three questions of Bertram on locally maximal sum-free sets

Anabanti, Chimere (2016) Three questions of Bertram on locally maximal sum-free sets. Technical Report. Birkbeck, University of London, London, UK.

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 [ SS [ SS−1 [ pS, where SS = {xy| x, y 2 S}, SS−1 = {xy−1| x, y 2 S} and pS = {x 2 G| x2 2 S}. Each set S in a finite group of odd order satisfies |pS| = |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: (i) Does S locally maximal sum-free in a finite abelian group imply |pS| � 2|S|? (ii) Does there exists a sequence of finite abelian groups G and locally maximal sum-free sets S � G such that |SS| |S| ! 1 as |G| ! 1? (iii) Does there exists a sequence of abelian groups G and locally maximal sum-free sets S � G such that |S| < c|G|1 2 as |G| ! 1, where c is a constant? In this paper, we answer question (i) in the negation, then (ii) and (iii) in affirmation.

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