The spectral study of a class of Moran measures in Rn

Let ${\left\{\right(A_k,D_k\left)\right\}}_{k=1}^{\infty }$ be a sequence of pairs, where $D_k$ is an integer vector set with ${\sup }_{d\in D_k}\parallel d\parallel <\infty$ and $A_k$ is an integer expansive matrix. Associated with the sequence ${\left\{\right(A_k,D_k\left)\right\}}_{k=1}^{\infty }$, Moran measure ${\mu }_{\left\{A_k\right\},\left\{D_k\right\}}$ is defined by${\mu }_{\left\{A_k\right\},\left\{D_k\right\}}={\delta }_{A_1^{-1}{D}_1}⁎{\delta }_{A_1^{-1}{A}_2^{-1}{D}_2}⁎\cdots .$ Assume that $\{x\in {(0,1)}^n:{\sum }_{d\in D_k}{e}^{2\pi i〈d,x〉}=0\}=q^{-1}{Z}^n\cap {[0,1)}^n﹨\left\{0\right\}$, we provide the necessary and sufficient conditions for $L^2\left({\mu }_{\left\{A_k\right\},\left\{D_k\right\}}\right)$ to have orthogonal exponential function bases under some metric conditions on $A_k$.