Around $P$-small subsets of groups

Abstract

A subset X of a group G is called P-small (almost P-small) if there exists an injective sequence (g n ) n∈ω in G such that the subsets (g n X) n∈ω are pairwise disjoint (g n X ∩ g m X is finite for all distinct n, m), and weakly P-small if, for every n ∈ ω, there exist g 0 , . . . , g n ∈ G such that the subsets g 0 X, . . . , g n X are pairwise disjoint. We generalize these notions and say that X is near P-small if, for every n ∈ ω, there exist g 0 , . . . , g n ∈ G such that g i X ∩ g j X is finite for all distinct i, j ∈ {0, . . . , n}. We study the relationships between near P-small subsets and known types of subsets of a group, and the behavior of near P-small subsets under the action of the combinatorial derivation and its inverse mapping.

Recent progress in subset combinatorics of groups

“…Each almost P -small subset can be partitioned into two P -small subsets [8]. Every countable Abelian group contains a near P -small subset which is neither weakly nor almost P -small [11].…”

Descriptive Complexity of the Sizes of Subsets of Groups

“…Each almost P -small subset can be partitioned into two P -small subsets [5]. Every countable Abelian group contains a near P -small subset which is neither weakly nor almost P -small [6].…”

“…2. The classification of subsets of a group by their size can be considered in much more general context of Asymptology (see [15]). In this context, large, thick and small subsets play the parts of dense, open and nowhere dense subsets of a uniform topological space.…”