Exponential riesz bases for two intervals and distributions

In this paper, we study hierarchically structured exponential Riesz bases for two disjoint intervals $$S_1, S_2 \subseteq [0, 1)$$ with unequal lengths. We show that there exist disjoint discrete sets $$\Lambda_1, \Lambda_2\subseteq \mathbb{Z}$$ such that the systems of exponentials with frequency sets $$\Lambda_1$$ and $$\Lambda_2$$ form Riesz bases for $$L^2(S_1)$$ and $$L^2(S_2)$$ respectively, and the system of exponentials with the frequency set $$\Lambda_1\cup\Lambda_2$$ constitutes a Riesz basis for $$L^2(S_1\cup S_2)$$ .