The spectrality of Cantor-Moran measure and Fuglede's Conjecture (2305.12135v2)

Abstract: Let ${(p_n, \mathcal{D}n, L_n)}$ be a sequence of Hadamard triples on $\mathbb{R}$. Suppose that the associated Cantor-Moran measure $$ \mu{{p_n,\mathcal{D}n}}=\delta{p_1^{-1}\mathcal{D}1}\ast\delta{(p_2p_1)^{-1}\mathcal{D}_2}\ast\cdots, $$ where $\sup_n{|p_n^{-1}d|:d\in \mathcal{D}n}<\infty$ and $\sup#\mathcal{D}_n<\infty$. 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 \textit{Double Points Condition Set} (\textit{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.