No abstract

We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that µ(A) = 0 for each left invariant Banach measure µ on G. It is also shown that every infinite group can be split into @0 scattered subsets.

Let G be a group and let Ä be a cardinal. A subset A of G is called left (right) Ä-large if there exists a subset F of G such that jF j < Ä and G D FA (G D AF ). We say that A is Ä-large if A is left and right Ä-large. It is known that every infinite group G can be partitioned into countably many @ 0 -large subsets. On the other hand, every amenable (in particular, Abelian) group G cannot be partitioned into more than @ 0 @ 0 -large subsets.We prove that every group G of cardinality Ä can be partitioned into Ä @ 1 -large subsets and every free group F Ä in the infinite alphabet Ä can be partitioned into Ä 4-large subsets but cannot be partitioned into three 3-large subsets.

We conjecture that every infinite group G can be partitioned into countably many cells G = n∈ω A n such that cov(A n A −1 n ) = |G| for each n ∈ ω. Here cov(A) = min{|X| : X ⊆ G, G = XA}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.2010 Mathematics Subject Classification. 03E05, 20B07, 20F69.

<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>