具有无理压缩比的卷积测度的非谱性研究

A Study of Non-Spectrality of Convolutional Measures with Irrational Contraction Ratio

摘要: 设${\mu }_{\rho ,D,\left\{n_k\right\}}$ 是由以下离散测度的无限卷积定义的Borel概率测度：${\mu }_{\rho ,D,\left\{n_k\right\}}={\delta }_{{\rho }^{n_1}D}\ast {\delta }_{{\rho }^{n_2}D}\ast {\delta }_{{\rho }^{n_3}D}\ast \cdots ,$ 其中$0<\rho <1$ ，$D\subset ℝ$ 是一有限集，${\left\{n_k\right\}}_{k=1}^{\infty }$ 是一个严格递增的正整数序列，且$\underset{k\ge 1}{\sup }\left\{n_{k+1}-n_k\right\}<\infty$ 。本文证明了：若$Z\left({\widehat{\delta }}_D\right)$ 包含在Lattice集中且对任意的$r>1$ ，无理数$\rho \cancel{\in }{\mathbb{Q}}^{1/r}:=\left\{\rho =u^{1/r}:0<u<1是一有理数\right\}$ ，那么${\mu }_{\rho ,D,\left\{n_k\right\}}$ 不是谱测度。

Abstract: Let ${\mu }_{\rho ,D,\left\{n_k\right\}}$ be a Borel probability measure defined by the following infinite convolution of discrete measures: ${\mu }_{\rho ,D,\left\{n_k\right\}}={\delta }_{{\rho }^{n_1}D}\ast {\delta }_{{\rho }^{n_2}D}\ast {\delta }_{{\rho }^{n_3}D}\ast \cdots ,$ where $0<\rho <1$ , $D\subset ℝ$ is a finite set, and${\left\{n_k\right\}}_{k=1}^{\infty }$ is a strictly increasing sequence of positive integers with $\underset{k\ge 1}{\sup }\left\{n_{k+1}-n_k\right\}<\infty$ . In this paper, we will show that if $Z\left({\widehat{\delta }}_D\right)$ is contained in a Lattice set and for any $r>1$ , the irrational number $\rho \cancel{\in }{\mathbb{Q}}^{1/r}:=\left\{\rho =u^{1/r}:0<u<1\text{is rational}\right\}$ , then ${\mu }_{\rho ,D,\left\{n_k\right\}}$ is not a spectral measure.