Spectrality of Moran-Type Bernoulli Convolutions

Let pn,dn∈Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_n,\ d_n\in {{\mathbb {Z}}}$$\end{document} be integers such that |pn|>|dn|>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|p_n|>|d_n|>0$$\end{document} and {dn}n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{d_n\}_{n\ge 1}$$\end{document} is bounded. It is proven that the Moran-type Bernoulli convolution μ:=δp1-1{0,d1}∗δp1-1p2-1{0,d2}∗⋯∗δp1-1⋯pn-1{0,dn}∗⋯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu :=\delta _{p_1^{-1}\{0,d_1\}}*\delta _{p_1^{-1}p_2^{-1} \{0,d_2\}}*\dots *\delta _{p_1^{-1}\dots p_n^{-1}\{0,d_n\}}*\dots \end{aligned}$$\end{document}is a spectral measure if and only if the numbers of factor 2 in the sequence {p1p2⋯pn2dn}n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big \{\frac{p_1p_2\dots p_n}{2d_n}\big \}_{n\ge 1}$$\end{document} are different from each other.