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

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 $F\subset G$.

💡 Research Summary

The paper investigates a new type of partition problem for countable infinite groups. The central objects are “thick” subsets of a group G (those that contain a left translate of every finite set) and their complement, “meager” subsets. A set X⊂G is called k‑meager if for every k‑element subset K⊂G the product KX is not thick. The main theorem asserts that for any natural number k and any countable infinite group G there exists a partition G=A∪B such that both A and B are k‑meager.

The proof hinges on the existence of a “syndetic submeasure” on G. A submeasure μ:℘(G)→