Spectrality of Moran measures with four-element digit sets

Let δE=1#E∑a∈Eδa denote the uniformly discrete probability measure on a finite set E. We prove that the infinite convolution (Moran measure)μb,{Dk}=δb−1D1⁎δb−2D2⁎⋯ admits an orthonormal basis of exponential provided that {Dk}k=1∞ is a uniformly bounded sequence of 4-digit spectral sets, b=2l+1q with q>1 an odd integer, and l sufficiently large (depends on Dk). We also give some examples to illustrate the result.