### Zero-sum subsets of decomposable sets in Abelian groups

T. Banakh, A. Ravsky

#### 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 $$\sum A+\sum B=0$$.

#### Keywords

decomposable set, abelian group, sum-set

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