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We prove that for every number k each countable infinite group $G$ admits apartition $G=A\cup B$ into two sets which are $k$-meager in the sense that forevery $k$-element subset $K\subset G$ the sets $KA$ and $KB$ are not thick. Theproof is based on the fact that $G$ possesses a syndetic submeasure, i.e., aleft-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 finitesubset $F\subset G$.

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