Partitions of groups into sparse subsets

Abstract

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 \(cov(A)=\min\{|X|: G = XA\}.\)