Thin subsets of groups

TL;DR

This paper investigates the structure of m-thin subsets in groups of various cardinalities, showing they can be partitioned into a finite number of 1-thin subsets, but also providing a counterexample in larger groups.

Contribution

It establishes a partitioning theorem for m-thin subsets in groups of countable and uncountable size, and constructs a counterexample for larger groups.

Findings

m-thin subsets of size ff_n can be partitioned into ff_{n+1} 1-thin subsets

Existence of a 2-thin subset in a group of size ff_\u03a9 that cannot be finitely partitioned into 1-thin subsets

Abstract

For a group $G$ and a natural number $m$, a subset $A$ of $G$ is called $m$-thin if, for each finite subset $F$ of $G$, there exists a finite subset $K$ of $G$ such that $|Fg\cap A|\leqslant m$ for every $g\in G\setminus K$. We show that each $m$-thin subset of a group $G$ of cardinality $\aleph_n$, $n= 0,1,...$ can be partitioned into $\leqslant m^{n+1}$ 1-thin subsets. On the other side, we construct a group $G$ of cardinality $\aleph_\omega$ and point out a 2-thin subset of $G$ which cannot be finitely partitioned into 1-thin subsets.