由容许对生成的无限卷积与随机测度的谱性及本征谱集
Spectrality and eigen sets of infinite convolutions and random measures generated by admissible pairs
AI总结：
本文通过无限卷积构造随机测度，引入t-等正族得到无限卷积为谱测度的充分条件，证明对应标准无限卷积非退化时几乎所有μ^n为谱随机测度，还刻画了数字集为连续集时的正整数谱本征值
AI中文摘要：
本文通过无限卷积构造了一类随机测度μ^n。给定容许对{(N_k,B_k)}_{k=1}^m及正整数序列bn={n_k}_{k=1}^∞，对每个ω∈Ω，记μ^n(ω)=δ_{N_{ω1}^{-n1}B_{ω1}} * δ_{N_{ω1}^{-n1}N_{ω2}^{-n2}B_{ω2}} * …。首先，证明映射μ^n:(ω,B)↦μ^n(ω)(B)是随机测度；其次，引入t-等正族的概念，并用其得到无限卷积为谱测度且具有指定谱本征值集的一般充分条件；接着，将谱性与谱本征值的概念推广到随机测度，证明在对应标准无限卷积对Ω中几乎所有ω均非退化的假设下，对Ω中几乎所有ω，测度μ^n均为谱随机测度，其谱本征谱集为ℰ_m={t∈ℕ_+:gcd(t,N_k)=1,1≤k≤m}；此外，对每个此类t，存在不可数多个谱Λ_ω⊂ℤ，使得tΛ_ω也是μ^n(ω)的谱；最后，针对每个数字集B_k均为连续集{0,1,…,b_k-1}的重要情形，完全刻画了μ^n(ω)的正整数谱本征值，证明其恰好为与每个b_k互素的整数。
英文摘要：
In this paper, we construct a class of random measures $μ^{\mathbf{n}}$ by infinite convolutions. Given admissible pairs $\{(N_{k}, B_{k})\}_{k=1}^{m}$ and a sequence $\bn=\{n_{k}\}_{k=1}^{\infty}$ of positive integers, for every $\bw\in Ω$, we write $μ^{\mathbf{n}}(\bw) = δ_{N_{ω_{1}}^{-n_{1}}B_{ω_{1}}} * δ_{N_{ω_{1}}^{-n_{1}}N_{ω_{2}}^{-n_{2}}B_{ω_{2}}} * \cdots$. First, we show that the mapping $μ^{\mathbf{n}}: (\bw, B) \mapsto μ^{\mathbf{n}}(\bw)(B)$ is a random measure. Next, we introduce the notion of a $t$-equi-positive family, and use it to obtain a general sufficient condition under which an infinite convolution is a spectral measure and possesses a specified set of spectral eigenvalues. We then extend the concepts of spectrality and spectral eigenvalues to random measures, and show that, under the assumption that the corresponding standard infinite convolution is non-degenerate for $\mathbb{P}$-a.e.\ $\boldsymbolω\inΩ$, the measures $μ^{\mathbf{n}}$ are spectral random measures for $\mathbb{P}$-a.e.\ $\boldsymbolω$, admitting the spectral eigen set \[ \mathcal{E}_m=\{t\in\mathbb{N}_+:\gcd(t,N_k)=1,\,1\le k\le m\}. \] Moreover, for each such $t$, there exist uncountably many spectra $Λ_{\boldsymbolω}\subset\mathbb{Z}$ with $tΛ_{\boldsymbolω}$ also a spectrum of $μ^{\mathbf{n}}(\boldsymbolω)$. Finally, for the important case where each digit set $B_k$ is a consecutive set $\{0,1,\dots,b_k-1\}$, we completely characterise the positive integer spectral eigenvalues of $μ^{\mathbf{n}}(\boldsymbolω)$, proving that they are exactly the integers coprime to every $b_k$.
