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

Let \(\{p_n\}_{n\ge 1}\) and \(\{ d_n\}_{n\ge 1}\) be two sequences of integers such that \(|p_n|>|d_n|>0\) and \(\{d_n\}_{n\ge 1}\) is bounded. It is proven by Deng and Li that the Moran-type Bernoulli convolution

$$\begin{aligned}\mu :=\delta _{p_1^{-1}\{0,d_1\}}*\delta _{p_1^{-1}p_2^{-1}\{0,d_2\}}*\dots *\delta _{p_1^{-1}\dots p_n^{-1}\{0,d_n\}}*\dots \end{aligned}$$

is a spectral measure if and only if the numbers of factor 2 in the sequence \(\big \{\frac{p_1p_2\dots p_n}{2d_n}\big \}_{n\ge 1}\) 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.