Syndetic submeasures and partitions of $G$-spaces and groups

TL;DR

This paper proves that every countable infinite group can be partitioned into two sets that are not thick with respect to any fixed finite subset, using the concept of syndetic submeasures.

Contribution

It introduces the use of syndetic submeasures to establish partitions of groups with specific non-thickness properties.

Findings

Existence of such partitions for all countable infinite groups.

Introduction of syndetic submeasures as a key tool.

Partitions are constructed to avoid thickness with respect to any finite subset.

Abstract

We prove that for every number k each countable infinite group $G$ admits a partition $G=A\cup B$ into two sets which are $k$-meager in the sense that for every $k$-element subset $K\subset G$ the sets $KA$ and $KB$ are not thick. The proof is based on the fact that $G$ possesses a syndetic submeasure, i.e., a left-invariant submeasure $\mu:\mathcal P(G)\to[0,1]$ such that for each $\epsilon > 1/|G|$ and subset $A\subset G$ with $\mu(A)<1$ there is a set $B\subset G\setminus A$ such that $\mu(B)<\epsilon$ and $FB=G$ for some finite subset $F\subset G$.