The spectrality of Cantor-Moran measure and Fuglede's Conjecture

Let $\{(p_n,{\mathcal{D}}_n,L_n)\}$ be a sequence of Hadamard triples on $\mathbb{R}$. Suppose that the associated Cantor-Moran measure $${\mu }_{\{p_n,{\mathcal{D}}_n\}}={\delta }_{p_1^{-1}{\mathcal{D}}_1}\ast {\delta }_{(p_2{p}_1{)}^{-1}{\mathcal{D}}_2}\ast \cdots ,$$ where ${\sup }_n\{\mathrm{\mid }{p}_n^{-1}d\mathrm{\mid }:d\in {\mathcal{D}}_n\}<\mathrm{\infty }$ and $\sup \mathrm{\#}{\mathcal{D}}_n<\mathrm{\infty }$. It has been observed that the spectrality of ${\mu }_{\{p_n,D_n\}}$ is determined by equi-positivity. A significant problem is what kind of Moran measures can satisfy this property. In this paper, we introduce the conception of $\mathit{\text{Double}}\mathit{\text{Points}}\mathit{\text{Condition}}\mathit{\text{Set}}$ ($\mathit{\text{DPCS}}$) to characterize the equi-positivity equivalently. As applications of our characterization, we show that all singularly continuous Cantor-Moran measures are spectral. For the absolutely continuous case, we study Fuglede’s Conjecture on Cantor-Moran set. We show that the equi-positivity of ${\mu }_{\{p_n,D_n\}}$ implies the tiling of its support, and the reverse direction holds under certain conditions.