Partitions of groups into sparse subsets

A subset $A$ of a group $G$ is called sparse if, for every infinite subset $X$ of $G$, there exists a finite subset $F\subset X$, such that $\bigcap_{x\in F} xA$ is finite. We denote by $\eta(G)$ the minimal cardinal such that $G$ can be partitioned in $\eta(G)$ sparse subsets. If $|G| > (\kappa^+)^{\aleph_0}$ then $\eta(G) > \kappa$, if $|G|\leqslant \kappa^+$ then $\eta(G) \leqslant \kappa$. We show also that $cov(A) \geqslant cf|G|$ for each sparse subset $A$ of an infinite group $G$, where $cov(A)=\min\{|X|: G = XA\}$.