$\mathbb{R}$上一类无穷卷积的谱
Spectrality of a class of infinite convolutions on $\mathbb{R}$
摘要：
给定一个整数$m\geq1$。设$Σ^{(m)}=\{1,2, \cdots, m\}^{\mathbb{N}}$是一个符号空间，$\{(b_{k},D_{k})\}_{k=1}^{m}:=\{(b_{k}, \{0,1,\cdots, p_{k}-1\}t_{k}) \}_{k=1}^{m}$是一个有限序列偶，其中整数$| b_{k}| $，$p_{k}\geq2$，$|t_{k}|\geq 1$和$ p_{k},t_{1},t_{2}, \cdots, t_{m}$是所有$1\leq k\leq m$的成对互质整数.本文证明了对于任意的无限字$σ=\left(σ_{n}\right)_{n=1}^{\infty}\inΣ^{(m)}$，无限卷积$$ μ_σ=δ_{b_{σ_{1}}^{-1} D_{σ_{1}}} * δ_{\left(b_{σ_{1}} b_{σ_{2}}\right)^{-1} D_{σ_{2}}} * δ_{\left(b_{σ_{1}} b_{σ_{2}} b_{σ_{3}}\right)^{-1}D_{σ_{3}}} * \cdots $$是谱测度当且仅当$p_{σ_n}\mid b_{σ_n}$对所有的$n\geq2$和$σ\notin \bigcup_{l=1}^\infty\prod_{l}$，其中$\prod_{l}=\{i_{1}i_{2}\cdots i_{l}j^{\infty}\inΣ^{(m)}: i_{l}\neq j, |b_{j}|=p_{j}, |t_{j}|\neq1\}$ .
英文摘要：
Given an integer $m\geq1$. Let $Σ^{(m)}=\{1,2, \cdots, m\}^{\mathbb{N}}$ 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 $σ=\left(σ_{n}\right)_{n=1}^{\infty}\inΣ^{(m)}$, the infinite convolution $$ μ_σ=δ_{b_{σ_{1}}^{-1} D_{σ_{1}}} * δ_{\left(b_{σ_{1}} b_{σ_{2}}\right)^{-1} D_{σ_{2}}} * δ_{\left(b_{σ_{1}} b_{σ_{2}} b_{σ_{3}}\right)^{-1}D_{σ_{3}}} * \cdots $$ is a spectral measure if and only if $p_{σ_n}\mid b_{σ_n}$ for all $n\geq2$ and $σ\notin \bigcup_{l=1}^\infty\prod_{l}$, where $\prod_{l}=\{i_{1}i_{2}\cdots i_{l}j^{\infty}\inΣ^{(m)}: i_{l}\neq j, |b_{j}|=p_{j}, |t_{j}|\neq1\}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
