Spectrality of a Class of Moran Measures on \(\mathbb {R}^n\)

Let \(\{M_k\}_{k=1}^\infty \) ${\{M_k\}}_{k=1}^{\infty }$ be a sequence of expansive matrices, and let \(\{D_k\}_{k=1}^\infty \) ${\{D_k\}}_{k=1}^{\infty }$ be a sequence of finite digit sets satisfying \(\mathcal {Z}_{D_k}^n=\mathcal {F}_{q_k}^n\) ${\mathcal{Z}}_{D_k}^n={\mathcal{F}}_{q_k}^n$ , where \(\mathcal {Z}_{D_k}^n=\{ x\in [0, 1)^n:\sum _{d\in D_k}{e^{2\pi i\langle d,x\rangle }}=0\}\) ${\mathcal{Z}}_{D_k}^n=\{x\in {[0,1)}^n:{\sum }_{d\in D_k}{e}^{2\pi i⟨d,x⟩}=0\}$ , \(\mathcal {F}_{q_k}^n=(\frac{\mathbb {Z}^n}{q_k}\cap [0, 1)^n)\setminus \{\textbf{0}\}\) ${\mathcal{F}}_{q_k}^n=(\frac{{\mathbb{Z}}^n}{q_k}\cap {[0,1)}^n)\backslash \{\mathbf{0}\}$ and the sequence \(\{q_k\}_{k=1}^\infty \) ${\{q_k\}}_{k=1}^{\infty }$ is bounded with \(q_k\ge 2\) $q_k\ge 2$ . In this paper, we show that the associated integral Moran measure \(\mu _{\{M_k\},\{D_k\}}\) ${\mu }_{\{M_k\},\{D_k\}}$ is a spectral measure if and only if \(\#D_k=q_k^n\) $\#D_k=q_k^n$ for all \(k\ge 1\) $k\ge 1$ and \(M_k\in M_n(q_k\mathbb {Z})\) $M_k\in M_n(q_k\mathbb{Z})$ for all \(k\ge 2\) $k\ge 2$ .