Zero-sum subsets of decomposable sets in Abelian groups

A subset $D$ of an Abelian group is $decomposable$ if $emptyset e Dsubset D+D$. In the paper we give partial answer to an open problem asking whether every finite decomposable subset $D$ of an Abelian group contains a non-empty subset $Zsubset D$ with $sum Z=0$. For every $ninmathbb N$ 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 $sum T e 0$ for any proper non-empty subset $Tsubset D$. On the other hand, we prove that every decomposable subset $Dsubsetmathbb R$ of cardinality $|D|le 7$ contains a non-empty subset $Zsubset D$ of cardinality $|Z|lefrac12|D|$ with $sum Z=0$. For every $ninmathbb N$ we present a subset $Dsubsetmathbb Z$ of cardinality $|D|=2n$ such that $sum Z=0$ for some subset $Zsubset D$ of cardinality $|Z|=n$ and $sum T e 0$ for any non-empty subset $Tsubset D$ 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 $sum A+sum B=0$.