Zero-sum subsets of decomposable sets in Abelian groups

T. Banakh ; A. Ravsky

Algebra and discrete mathematics, Tome 30 (2020) no. 1, pp. 15-25.

Résumé

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 $|T|$. Also we prove that every finite decomposable subset $D$ of an Abelian group contains two non-empty subsets $A$, $B$ such that $\sum A+\sum B=0$.

Détail

Keywords: abelian group, sum-set.
Mots-clés : decomposable set

Bibliographie

[1] Aryabhata et al., A zero sum subset of a sum-full set, Mathematics StackExchange https://math.stackexchange.com/q/2418

[2] Hsieh-Chih Chang, A partial answer to a problem of Zaimi, MathOverflow https://mathoverflow.net/a/16871

[3] Gjergji Zaimi et al., Existence of a zero-sum subset, MathOverflow https://mathoverflow.net/q/16857