Thin systems of generators of groups

A subset $T$ of a group $G$ with the identity $e$ is called $k$-thin $(k\in{\mathbb N})$ if $|A\cap gA|\leqslant k$, $|A\cap Ag|\leqslant k$ for every $g\in G$, $g\ne e$. We show that every infinite group $G$ can be generated by some 2-thin subset. Moreover, if $G$ is either Abelian or a torsion group without elements of order 2, then there exists a 1-thin system of generators of $G$. For every infinite group $G$, there exist a 2-thin subset $X$ such that $G=XX^{-1}\cup X^{-1}X$, and a 4-thin subset $Y$ such that $G=YY^{-1}$.