@article{Protasov_Slobodianiuk_2015, title={The dynamical look at the subsets of a group}, volume={16}, url={https://polipapers.upv.es/index.php/AGT/article/view/3584}, DOI={10.4995/agt.2015.3584}, abstractNote={<p>We consider the action of a group $G$ on the family $\mathcal{P}(G)$ of all subsets of $G$ by the right shifts $A\mapsto Ag$ and give the dynamical characterizations of thin, $n$-thin, sparse and scattered subsets.</p><p>For $n\in\mathbb{N}$, a subset $A$ of a group $G$ is called $n$-thin if $g_0A\cap\dots\cap g_nA$ is finite for all distinct $g_0,\dots,g_n\in G$.<br />Each $n$-thin subset of a group of cardinality $\aleph_0$ can be partitioned into $n$ $1$-thin subsets but there is a $2$-thin subset in some Abelian group of cardinality $\aleph_2$ which cannot be partitioned into two $1$-thin subsets. We eliminate the gap between $\aleph_0$ and $\aleph_2$ proving that each $n$-thin subset of an Abelian group of cardinality $\aleph_1$ can be partitioned into $n$ $1$-thin subsets.</p>}, number={2}, journal={Applied General Topology}, author={Protasov, Igor V. and Slobodianiuk, Sergii}, year={2015}, month={Oct.}, pages={217–224} }