Spectrality of a class of infinite convolutions

For a finite set D⊂ZD⊂Z and an integer b≥2b≥2, we say that (b,D)(b,D) is compatible with C⊂ZC⊂Z if [e−2πidc/b]d∈D,c∈C[e−2πidc/b]d∈D,c∈C is a Hadamard matrix. Let δE=1#E∑a∈Eδa denote the uniformly discrete probability measure on E  . We prove that the class of infinite convolution (Moran measure) μb,{Dk}=δb−1D1⁎δb−2D2⁎⋯μb,{Dk}=δb−1D1⁎δb−2D2⁎⋯ is a spectral measure provided that there is a common C⊂Z+C⊂Z+ compatible to all the (b,Dk)(b,Dk) and C+C⊆{0,1,…,b−1}C+C⊆{0,1,…,b−1}. We also give some examples to illustrate the hypotheses and results, in particular, the last condition on CC is essential.