Spectral property of a class of Moran measures on R

Let b≥2 be a positive integer. Let D be a finite subset of Z and {nk}k=1∞⊆N be a sequence of strictly increasing numbers. A Moran measure μb,D,{nk} is a Borel probability measure generated by the Moran iterated function system (Moran IFS) {fk,d(x)=bnk−1−nk(x+d):d∈D,k∈N,n0=0}. In this paper we study one of the basic problems in Fourier analysis associated with μb,D,{nk}. More precisely, we give some conditions under which the measure μb,D,{nk} is a spectral measure, i.e., there exists a discrete subset Λ⊆R such that E(Λ)={e2πiλx:λ∈Λ} is an orthonormal basis for L2(μb,D,{nk}).