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 $(g_{n})_{n\in\omega}$ in $G$ such that the subsets $(g_{n}X)_{n\in\omega}$  are pairwise disjoint ($g_{n}X\cap g_{m}X$ is finite for all distinct $n,m$), and weakly $P$-small if, for every $n\in\omega$, there exist $g_{0}, \ldots ,g_{n}\in G$ 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 $n\in\omega$, there exist $g_{0}, \ldots ,g_{n}\in G$ such that $g_{i}X\cap g_{j}X$ is finite for all distinct $i,j \in\{0,\ldots, n\}$. 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.