Scattered Subsets of Groups

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.

Descriptive Complexity of the Sizes of Subsets of Groups

“…3. The following type of subsets of a group arised in Asymptology [11]. A subset A of a group G is called scattered if A has no subsets coarsely equivalent to the Cantor macrocube.…”

“…Each sparse subset is scattered, each scattered subset is small and the set of all scattered subsets of G is an ideal in P G . By Theorem 1 [11], a subset A of a group G is scattered if and only if A contains no piecewise shifted F P -sets.…”

The dynamical look at the subsets of a group

“…Remark 4.7. By [1,Theorem 3], every infinite group G can be partitioned into ℵ 0 scattered subsets. For a group G we denote by µ(G) and η(G) the minimal cardinalitis of partitions of G into thin and sparse subsets respectivly.…”

“…Scattered subsets were introduced in [1] as asymptotic counterparts of scattered topological spaces.…”

Partitions of Groups

“…The combinatorial derivation was introduced in [33] and studied in [12], [34], [38]. The results of this sections from [5], [38], [39]. For ultracompanions of subsets of balleans see [8].…”