Spectrality of Moran-type measures with staggered contraction ratios

Abstract: Consider a Moran-type iterated function system (IFS) ( {\phi_{k,d}}{d\in D{2p_k}, k\geq 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}^\infty ) and ( {p_k}{k=1}^\infty ) satisfying ( b_k \geq 2p_k \geq 2 ), and digit sets ( D_{2p_k} = {0, 1, \ldots, 2p_k - 1} ) for all ( k \geq 1 ). We first prove that this IFS uniquely generates a Borel probability measure ( \mu ). Furthermore, under the divisibility constraints [ p_2 \mid b_2, \quad 2 \mid b_2, \quad \text{and} \quad 2p_k \mid b_k \ \text{for} \ k \geq 3, ] with ({b_k}_{k=1}^\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.