Riesz bases of exponentials on multiband spectra

Let $S$ be the union of finitely many disjoint intervals on $\mathbb{R}$. Suppose that there are two real numbers $\alpha , \beta$ such that the length of each interval belongs to $\mathbb{Z} \alpha + \mathbb{Z}\beta$. We use quasicrystals to construct a discrete set $\Lambda \subset \mathbb{R}$ such that the system of exponentials $\{\exp 2 \pi i \lambda x, \, \lambda \in \Lambda \}$ is a Riesz basis in the space $L^2(S)$.