Spectrality of a class of infinite convolutions on $\mathbb{R}$

Abstract

Given an integer $m\ge 1$. 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 $|b_k|$, $p_k\ge 2$, $|t_k|\ge 1$ and $p_k,t_1,t_2,\cdots ,t_m$ are pairwise coprime integers for all $1\le k\le m$. In this paper, we show that for any infinite word $\sigma ={\left({\sigma }_n\right)}_{n=1}^{\infty }\in {\Sigma }^{(m)}$, the infinite convolution $${\mu }_{\sigma }={\delta }_{b_{{\sigma }_1}^{-1}{D}_{{\sigma }_1}}\ast {\delta }_{{\left(b_{{\sigma }_1}{b}_{{\sigma }_2}\right)}^{-1}{D}_{{\sigma }_2}}\ast {\delta }_{{\left(b_{{\sigma }_1}{b}_{{\sigma }_2}{b}_{{\sigma }_3}\right)}^{-1}{D}_{{\sigma }_3}}\ast \cdots$$ is a spectral measure if and only if $p_{{\sigma }_n}\mid b_{{\sigma }_n}$ for all $n\ge 2$ and $\sigma \notin {\bigcup }_{l=1}^{\infty }{\prod }_l$, where ${\prod }_l=\{i_1{i}_2\cdots i_l{j}^{\infty }\in {\Sigma }^{(m)}:i_l\ne j,|b_j|=p_j,|t_j|\ne 1\}$.