A Note on the Spectrality of Moran-Type Bernoulli Convolutions by Deng and Li

Let \(\{p_n\}_{n\ge 1}\) ${\{p_n\}}_{n\ge 1}$ and \(\{ d_n\}_{n\ge 1}\) ${\{d_n\}}_{n\ge 1}$ be two sequences of integers such that \(|p_n|>|d_n|>0\) $|p_n|>|d_n|>0$ and \(\{d_n\}_{n\ge 1}\) ${\{d_n\}}_{n\ge 1}$ is bounded. It is proven by Deng and Li that the Moran-type Bernoulli convolution \(\begin{aligned}\mu :=\delta _{p_1^{-1}\{0,d_1\}}*\delta _{p_1^{-1}p_2^{-1}\{0,d_2\}}*\dots *\delta _{p_1^{-1}\dots p_n^{-1}\{0,d_n\}}*\dots \end{aligned}\) $$\begin{array}{c}\hfill \mu :={\delta }_{p_1^{-1}\{0,d_1\}}\ast {\delta }_{p_1^{-1}{p}_2^{-1}\{0,d_2\}}\ast \cdots \ast {\delta }_{p_1^{-1}\cdots p_n^{-1}\{0,d_n\}}\ast \cdots \end{array}$$ is a spectral measure if and only if the numbers of factor 2 in the sequence \(\big \{\frac{p_1p_2\dots p_n}{2d_n}\big \}_{n\ge 1}\) $\{\frac{p_1{p}_2\cdots p_n}{2d_n}{\}}_{n\ge 1}$ are different from each other. Unfortunately, there is a gap in the proof of the sufficiency. Here we give a new proof to close the gap.