Zero-sum subsets of decomposable sets in Abelian groups

Abstract

A subset \(D\) of an abelian group is decomposable if \( \emptyset\ne D\subset D+D\). In the paper we give partial answers to an open problem asking whether every finite decomposable subset \(D\) of an abelian group contains a non-empty subset \(Z\subset D\) with \(\sum Z=0\). For every \(n\in\mathbb{N}\) we present a decomposable subset \(D\) of cardinality \(|D|=n\) in the cyclic group of order \(2^n-1\) such that \(\sum D=0\), but \(\sum T\ne 0\) for any proper non-empty subset \(T\subset D\). On the other hand, we prove that every decomposable subset \(D\subset\mathbb{R}\) of cardinality \(|D|\le 7\) contains a non-empty subset \(T\subset D\) of cardinality \(|Z|\le\frac12|D|\) with \(\sum Z=0\). For every \(n\in\mathbb{N}\) we present a subset \(D\subset\mathbb{Z}\) of cardinality \(|D|=2n\) such that \(\sum Z=0\) for some subset \(Z\subset D\) of cardinality \(|Z|=n\) and \(\sum T\ne 0\) for any non-empty subset \(T\subset D\) of cardinality \(|T|<n=\frac12|D|\). Also we prove that every finite decomposable subset \(D\) of an Abelian group contains two non-empty subsets \(A,B\) such that \(\sum A+\sum B=0\).