  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 $F\subset G$.
