# Do syndetic sets on amenable semigroups have positive upper density?

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

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 $D^*(S) = \limsup_{n \to \infty} \frac{2n + 2}{2n^2 + 2n + 2} = 0$.