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 $\prod_{ l}=\{ i_{ 1}i_{ 2}\cdots i_{ l}j^{ \infty}\in\Sigma^{ (m)}: i_{ l}\neq j, |b_{ j}|=p_{ j}, |t_{ j}|\neq1\}$.
Categories: math.CA math.DS