Zero-sum subsets of decomposable sets in Abelian groups

Abstract

A subset $D$ of an Abelian group is $decomposable$ if $\varnothing \ne D\subset D+D$. 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 $Z\subset D$ 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 $Z\subset D$ of cardinality $|Z|\le \frac{1}{2}|D|$ 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 $|T|<n=\frac{1}{2}|D|$. Also we prove that every finite decomposable subset $D$ of an Abelian group contains two non-empty subsets $A,B$ such that $\sum A+\sum B=0$.

8 pages