Let $\mathbb{G}$ be a discrete amenable semigroup, and $\left\{ F\_{n}\right\} $
a Folner sequence. 


For $S\subset \mathbb{G}$ define the upper density as $D^{\ast
}(S)=\limsup\_{n\rightarrow \infty }\frac{\left\vert S\cap F\_{n}\right\vert }{%
\left\vert F\_{n}\right\vert }.$


Suppose the exists $m\geq 0$ such that for every $g\in \mathbb{G}$, we have
that $gF\_{m}\cap S\neq \varnothing .$ 


Is $D^{\ast }(S)>0?$



Do syndetic sets on amenable semigroups have positive upper density?
This is true if $F\_n$ is a *right* Følner sequence, i.e., $\frac{|F\_n \Delta F\_ng |}{|F\_n|} \to 0$ for all $g \in G$. Indeed, if $F \subset G$ is finite, then the condition $g F \cap S \not= \emptyset$, for every $g \in G$, is equivalent to the condition $S F^{-1} = G$. Hence $D^\*(S) = \frac{1}{|F|} \sum\_{f \in F} D^\* (Sf^{-1}) \geq \frac{1}{|F|} D^\*(S F^{-1}) = \frac{1}{|F|}$.


This can fail however if we consider only *left* Følner sequences. A counter-example is given by letting $G$ be the infinite dihedral group $\mathbb Z \rtimes^\alpha (\mathbb Z / 2 \mathbb Z)$, where $\alpha$ implements the flip automorphism on $\mathbb Z$. If we set $S = \mathbb N \cup (-\mathbb N)\alpha \subset G$, and $F = \{ 0, \alpha \}$, then we have $g F \cap S \not= \emptyset$ for all $g \in G$. However, considering $F\_n = [-n^2, n] \cup [- n, n^2]\alpha$ we have that $F\_n$ is a left Følner sequence such that $D^\*(S) = \limsup\_{n \to \infty} \frac{2n + 2}{2n^2 + 2n + 2} = 0$.


