Spectrality of a class of infinite convolutions on $\mathbb{R}$ (2212.00340v1)

Abstract: Given an integer $m\geq1$. Let $\Sigma^{(m)}={1,2, \cdots, m}^{\mathbb{N}}$ be a symbolic space, and let ${(b_{k},D_{k})}{k=1}^{m}:={(b{k}, {0,1,\cdots, p_{k}-1}t_{k}) }{k=1}^{m}$ be a finite sequence pairs, where integers $| b{k}| $, $p_{k}\geq2$, $|t_{k}|\geq 1$ and $ p_{k},t_{1},t_{2}, \cdots, t_{m}$ are pairwise coprime integers for all $1\leq k\leq m$. In this paper, we show that for any infinite word $\sigma=\left(\sigma_{n}\right){n=1}^{\infty}\in\Sigma^{(m)}$, the infinite convolution $$ \mu{\sigma}=\delta_{b_{\sigma_{1}}^{-1} D_{\sigma_{1}}} * \delta_{\left(b_{\sigma_{1}} b_{\sigma_{2}}\right)^{-1} D_{\sigma_{2}}} * \delta_{\left(b_{\sigma_{1}} b_{\sigma_{2}} b_{\sigma_{3}}\right)^{-1}D_{\sigma_{3}}} * \cdots $$ is a spectral measure if and only if $p_{\sigma_n}\mid b_{\sigma_n}$ for all $n\geq2$ and $\sigma\notin \bigcup_{l=1}^\infty\prod_{l}$, where $\prod_{l}={i_{1}i_{2}\cdots i_{l}j^{\infty}\in\Sigma^{(m)}: i_{l}\neq j, |b_{j}|=p_{j}, |t_{j}|\neq1}$.