Integer tile and Spectrality of Cantor-Moran measures with equidifferent digit sets

Let ${\left\{b_k\right\}}_{k=1}^{\mathrm{\infty }}$ be a sequence of integers with $\mathrm{\mid }{b}_k\mathrm{\mid }\ge 2$ and ${\left\{D_k\right\}}_{k=1}^{\mathrm{\infty }}$ be a sequence of equidifferent digit sets with $D_k=\{0,1,\cdots ,N-1\}{t}_k,$ where $N\ge 2$ is a prime number and $\{t_k{\}}_{k=1}^{\mathrm{\infty }}$ is bounded. In this paper, we study the existence of the Cantor-Moran measure ${\mu }_{\{b_k\},\{D_k\}}$ and show that $${\mathbf{D}}_k:=D_k\oplus b_k{D}_{k-1}\oplus b_k{b}_{k-1}{D}_{k-2}\oplus \cdots \oplus b_k{b}_{k-1}\cdots b_2{D}_1$$ is an integer tile for all $k\in {\mathbb{N}}^{+}$ if and only if ${\mathbf{s}}_i\mathrm{\ne }{\mathbf{s}}_j$ for all $i\mathrm{\ne }j\in {\mathbb{N}}^{+}$, where ${\mathbf{s}}_i$ is defined as the numbers of factor $N$ in $\frac{b_1{b}_2\cdots b_i}{N{t}_i}$. Moreover, we prove that ${\mathbf{D}}_k$ being an integer tile for all $k\in {\mathbb{N}}^{+}$ is a necessary condition for the Cantor-Moran measure to be a spectral measure, and we provide an example to demonstrate that it cannot become a sufficient condition. Furthermore, under some additional assumptions, we establish that the Cantor-Moran measure to be a spectral measure is equivalent to ${\mathbf{D}}_k$ being an integer tile for all $k\in {\mathbb{N}}^{+}$.