$S-S$ is syndetic set if $S$ has positive upper density, in the case of group action Let $G$ be a discrete group.  A sequence $\mathcal{F}=\{F_n\}$ is called a $Folner$ sequence if $\frac{|gF_n\Delta F_n|}{|F_n|}\to 0$ as $n\to \infty$ for every $g\in G$. $F_n$ is a finite subset of $G$ for all $n$.
$S\subseteq G$ is called a syndetic set if there is a finite set $F\subseteq G$ such that $G=FS$.
We can define respectively the upper density and lower density of $S$ with respect to $\mathcal{F}$ by 
\begin{equation*}
  \overline{d}_{\mathcal{F}}(S)= \limsup_{n\to \infty}\frac{|S\cap F_n|}{|F_n|}, \text{  and }  \underline{d}_{\mathcal{F}}(S)= \liminf_{n\to \infty}\frac{|S\cap F_n|}{|F_n|}
\end{equation*}
Let $S\subseteq G$ be a subset of $G$ of positive upper density. In the case of $G=\mathbb{Z}$, it is known that $SS^{-1}=S-S$ is a syndetic set. 
Let $G$ be an infinite discrete group. Is it true that $S^2$ or $SS^{-1}$ is syndetic?
Please help me to know it.
 A: First, for $S^2$ the claim fails even for $G=\Bbb Z$,  $F_n=\{1,\dots,n\}$  for each $n$, and $S=\Bbb N$. Then $\overline{d}_{\mathcal F}(S)=1$, but $S^2$ is not syndetic. Next, by the Følner criterion [Pat, 4.10], a group $G$ admits a Følner sequence iff $G$ is countable and amenable, see [Ban1, the end of p.2] for the definitions of amenable group.  Now let $G$ be a countable amenable group, $\mathcal F$ be a Følner sequence for $G$, and $S$ be a subset of $G$ such that $\overline{d}_{\mathcal F}(S)>0$. Then the upper Banach density $d^*(S)=\sup\{\overline{d}_{\mathcal H}(S): \mathcal H\mbox{ is a Følner sequence }\}>0$. By [Theorem 5.1, Ban1] the right Solecki density $\sigma^R(S)= d^*(S)>0$. By [Proposition 12.2, Ban1] there exists a subset $F$ of $G$ such that $|F|<1/ \sigma^R(S)$    and $G=FSS^{-1}$.
References
[Ban1] Taras Banakh, The Solecki submeasures and densities on groups.
[Ban2] Taras Banakh, Extremal densities and measures on groups and $G$-spaces and their combinatorial applications.
[BPS] Taras Banakh, Igor Protasov, Sergiy Slobodianiuk, Densities, submeasures, and partitions of groups,  Algebra Discr. Math. 17:2 (2014), 193–221.
[Pat] A. Paterson, Amenability, Math. Surveys and Monographs. 29, Amer. Math. Soc. Providece, RI, 1988.
