A note on the spectrality of Moran-type Bernoulli convolutions by Deng and Li (Q6587093)

Let \(\{p_n\}_{n\geq 1}\) and \(\{d_n\}_{n\geq 1}\) be two sequences of integers satisfying \(|p_n|\geq 2\), \(|d_n|\geq 1\) and\N\[\N\sum_{n=1}^{\infty} |p_1^{-1} p_2^{-1}\ldots p_n^{-1} d_n| < +\infty.\N\]\NThe weak limit of the following convolutions is called a Moran-type Bernoulli convolution\N\[\N\mu_n = \delta_{p_1^{-1}D_1} \ast \delta_{p_1^{-1}p_2^{-1}D_2}\ast\ldots\ast \delta_{p_1^{-1}p_2^{-1}\ldots p_n^{-1}D_n}.\N\]\NWe denote it by\N\[\N\mu = \delta_{p_1^{-1}D_1} \ast \delta_{p_1^{-1}p_2^{-1}D_2}\ast\ldots\ast \delta_{p_1^{-1}p_2^{-1}\ldots p_n^{-1}D_n}\ldots. \tag{1}\N\]\N\NIn [\textit{Q.-R. Deng} and \textit{M.-T. Li}, ibid. 46, No. 4, Paper No. 136, 19 p. (2023; Zbl 1518.42014)] the following result was obtained.\N\NTheorem 1. For the measure \(\mu\) defined by (1) with \(|p_n|>|d_n|\) for all \(n\geq 2\), assume that the sequence \(\{|d_n|\}_{n\geq 1}\) is bounded. Then, \(\mu\) is a spectral measure if and only if \(k_j \neq k_i\) for all \(j > i\geq 1\), where\N\[\Nk_n = v_2 \big(p_1 p_2\ldots p_n /(2d_n)\big)= v_2(p_1 p_2 \ldots p_n) - v_2(2d_n),\quad n = 1, 2, 3,\ldots.\N\]\N\NHowever, the proof of sufficiency in Theorem 1 contained a gap. In this paper, the authors reprove the sufficiency of Theorem 1.