Let $ (M, \omega) $ be a symplectic manifold. The de Rham class of $\omega$ induces a homomorphism $[\omega]: H_1(M) \to \mathbb{R}$, whose image $\Gamma_{\omega} \subseteq \mathbb{R}$ is called the group of periods of $\omega$. In the classical theory of prequantization one assumes that $\omega$ is integral, i.e. $\Gamma_{\omega} \subseteq \mathbb{Z}$. More generally, one can construct a prequantization principal $\mathbb{R} / \Gamma_\omega$ bundle if $\Gamma_\omega$ is a discrete subgroup of $\mathbb{R}$. The only other case is that the group of periods is dense.

Question: What are known examples of closed symplectic manifolds with a dense group of periods?