I've read several times the assertion "Reeb periods are countable" but couldn't find a reference for it.
Let $(M,\xi)$ be a contact manifold and $\alpha$ a contact form. The Reeb vector field $\mathcal{R}$ is the unique vector field such that $i_{\mathcal{R}}d\alpha=0$ and $i_{\mathcal{R}}\alpha=1$.
Let $x:\mathbb{R}\to M$ be an orbit of $\mathcal{R}$, that is $\dot{x}(t)=\mathcal{R}(x(t))$. As usual we say that $x$ has period $T$ if $x(t+T)=x(t)$. Let's denote $$ sp(\mathcal{R})=\{T\colon \exists \text{ periodic orbit of } \mathcal{R} \text{ of period }T\}. $$
I'm interested in a proof of the following lemma.
Lemma. If $M$ is compact, then $sp(\mathcal{R})$ is countable.
Furthermore, I would be interested if the compact condition can be lifted.