Spectrality of Moran-type measures with staggered contraction ratios

Consider a Moran-type iterated function system (IFS) $\{{\phi }_{k,d}{\}}_{d\in D_{2p_k},k\ge 1}$, where each contraction map is defined as $${\phi }_{k,d}(x)=(-1{)}^d{b}_k^{-1}(x+d),$$ with integer sequences $\{b_k{\}}_{k=1}^{\mathrm{\infty }}$ and $\{p_k{\}}_{k=1}^{\mathrm{\infty }}$ satisfying $b_k\ge 2p_k\ge 2$, and digit sets $D_{2p_k}=\{0,1,\dots ,2p_k-1\}$ for all $k\ge 1$. We first prove that this IFS uniquely generates a Borel probability measure $\mu$. Furthermore, under the divisibility constraints $$p_2\mid b_2,2\mid b_2,\text{and}2p_k\mid b_k\text{ for }k\ge 3,$$ with $\{b_k{\}}_{k=1}^{\mathrm{\infty }}$ bounded, we prove that $\mu$ is a spectral measure, that is, $L^2(\mu )$ admits an orthogonal basis of exponentials. To fully characterize the spectral properties, we introduce a multi-stage decomposition strategy for spectrums. By imposing the additional hypothesis that all parameters $p_k$ are even, we establish a complete characterization of $\mu$’s spectrality. This result unifies and extends the frameworks proposed in \cite{An-He2014, Deng2022, Wu2024}, providing a generalized criterion for such measures.