Weak convergence and spectrality of infinite convolutions

Let {Ak}k=1∞ be a sequence of finite subsets of Rd satisfying that #Ak≥2 for all integers k≥1. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolutionν=δA1⁎δA2⁎⋯⁎δAn⁎⋯, where all sets Ak⊆R+d and δA=1#A∑a∈Aδa. Then we study the spectrality of a class of infinite convolutions generated by Hadamard triples in R and construct a class of singular spectral measures without compact support. Finally we show that such measures are abundant, and the dimension of their supports has the intermediate-value property.