Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Zero-sum subsets of decomposable sets in Abelian groups   A subset $D$ of an Abelian group is $decomposable$ if $\emptyset\ne D\subset
D+D$. 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 $|Z|\le\frac12|D|$ 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 $\sum A+\sum
B=0$.

8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol