The existence of Fourier basis for some Moran measures

Let {bn}n=1∞ be a sequence of integers bigger than 1 and let {Dn}n=1∞ be a sequence of digit sets in Z where Dn={0,rn,2rn,⋯,(qn−1)rn}. The family of functions {fn,d(x)=bn−1(x+d):d∈Dn}n=1∞ is called a Moran iterated function system (IFS). In this paper we prove that the associated Moran measure generated by an infinite convolution of atomic measures with equal distributionμ{bn},{Dn}=δb1−1D1⁎δ(b1b2)−1D2⁎⋯⁎δ(b1b2⋯bn)−1Dn⁎⋯ is a spectral measure if rnqn|bn.