Does this property characterizes amenability or there are examples of non-amenable groups satisfying it?


Let $G$ be finitely generated group.


Property:


There exists $C<1$ such that for every $S\subset G$ - finite set, there exists $F \subset G$ - finite, such that 


$|sF \Delta F|\leq C\cdot |F|$ for every $s\in S$


Currently I don't see if it is possible to rebuilt sets $F$ in order to construct Folner sequence for $G$.


The question is related to [Amenability of groups](https://mathoverflow.net/questions/44843/amenability-of-groups)



a question on Folner sets
If I am not mistaken the group should be amenable by the following arguments.


As was explained to me by Jesse Peterson, if $\pi:G\rightarrow B(H)$ is a representation of $G$ and there exists unit vector $\xi$ such that for every $g\in G$: $\|\pi(g)\xi-\xi\|\leq C$ then $\pi$ has an invariant vector.


Let $\xi\_F=\frac{1}{\sqrt{F}}1\_{F}$, then from the property we have $||\lambda(s)\xi\_F-\xi\_{F}||\leq C$ for every $s\in S$. Then taking $S\_i$ as an increasing sequence such that $\cup S\_i=G$ and ultralimit of $\lambda$, denote $\lambda\_{\omega}:G\rightarrow B(l\_2(G)^{\omega})$, we have the existence of a unit vector $\xi$ such that $||\lambda\_{\omega}(g)\xi-\xi||\leq C$ for every $g\in G$. Thus $\lambda\_{\omega}$ has an invariant vector, which implies that $\lambda$ has a sequence of almost invariant vectors. The last one is equivalent to amenability. 


