The dynamical look at the subsets of a group

Abstract

<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>

Orbitally discrete coarse spaces

“…Is X orbitally discrete? For finite partitions of an n-thin space into discrete subspaces, see [5], [14], [17], [1, Section 6]. 2.…”

Set-Theoretical Problems in Asymptology

“…The action of the group G extends to a continuous action of G on the Stone-Čech compactification βX of the discrete space X. The remainder X * = βX \ X is an invariant subset of the action, so we obtain a dynamical system (X * , G) and can study the interplay between properties of the coarse structure E G and the properties of the dynamical system (X * , G), see the papers [19], [24], [30] for more information on this topic. By a dynamical system we understand a compact Hausdorff space K endowed with the continuous action of some group G.…”

“…Given a countable subgroup G ⊆ S ω , consider the finitary coarse structure E G on ω induced by the action of the group G. By analogy with Theorem 4.1 [19], it can be shown that for any P -point p ∈ ω * the set p coincides with the closure Gp of the orbit of p under the action of the group G on ω * . Problem 5.8 ( [19]). Is a free ultrafilter p on ω a P -point if p = Gp for any countable subgroup G ⊆ S ω ?…”

Recent progress in subset combinatorics of groups

“…Let G be an arbitrary infinite group. In [15], we constructed two injective sequences (x n ) n∈ω , (y n ) n∈ω in G such the set {x n y m : 0 ≤ n ≤ m < ω} is scattered. By Theorem 4.3(ii), this subset is not sparse.…”