Zero-sum subsets of decomposable sets in Abelian groups

A subset $D$ of an abelian group is decomposable if $\varnothing\ne D\subset D+D$. 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 $\sum A+\sum B=0$.