On the spectrality of a class of Moran measures

In this paper, we study the spectrality of a class of Moran measures μP,D on R generated by \(pₙ,Dₙ)\ₙ₌₁∞, where P=ₙ\ₙ₌₁∞ is a sequence of positive integers with pₙ>1 and D=ₙ\ₙ₌₁∞ is a sequence of digit sets of N with the cardinality #Dₙ∈ \2,3,Nₙ\. We find a countable set Λ such that the set -2π i λ x|λ∈Λ\ is a orthonormal basis of L²(μP,D) under some conditions. As an application, we show that when μP,D is absolutely continuous, μP,D not only is a spectral measure, but also its support set tiles R with Z.