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 $μ:\mathcal P(G)\to[0,1]$ such that for each $ε> 1/|G|$ and subset $A\subset G$ with $μ(A)<1$ there is a set $B\subset G\setminus A$ such that $μ(B)<ε$ and $FB=G$ for some finite subset $F\subset G$.

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