Spectrality of random convolutions generated by finitely many Hadamard triples

TL;DR

We address the spectrality of random convolutions generated by finitely many Hadamard triples in $\mathbb{R}$, introducing an equi-positivity/admissibility framework to guarantee spectrality of weak limits. A general result shows that if a tail sequence is equi-positive, the limit measure has a spectrum in $\mathbb{Z}$; under the condition $\gcd(B_j - B_j)=1$, every infinite convolution $\mu_{\omega,\{n_k\}}$ is spectral, with a constructed spectrum. The work provides a practical criterion for ensuring spectrality in non-compact settings and broadens the class of fractal-like random convolutions known to be spectral. These contributions advance understanding of spectral fractal measures and offer constructive methods for generating spectral random convolutions.

Abstract

Let $\{(N_j, B_j, L_j): 1 \le j \le m\}$ be finitely many Hadamard triples in $\mathbb{R}$. Given a sequence of positive integers $\{n_k\}_{k=1}^\infty$ and $ω=(ω_k)_{k=1}^\infty \in \{1,2,\cdots, m\}^\mathbb{N}$, let $μ_{ω,\{n_k\}}$ be the infinite convolution given by $$μ_{ω,\{n_k\}} = δ_{N_{ω_1}^{-n_1} B_{ω_1}} * δ_{N_{ω_1}^{-n_1} N_{ω_2}^{-n_2} B_{ω_2}} * \cdots * δ_{N_{ω_1}^{-n_1} N_{ω_2}^{-n_2} \cdots N_{ω_k}^{-n_k} B_{ω_k} }* \cdots. $$ In order to study the spectrality of $μ_{ω,\{ n_k\}}$, we first show the spectrality of general infinite convolutions generated by Hadamard triples under the equi-positivity condition. Then by using the integral periodic zero set of Fourier transform we show that if $\mathrm{gcd}(B_j - B_j)=1$ for $1 \le j \le m$, then all infinite convolutions $μ_{ω,\{n_k\}}$ are spectral measures. This implies that we may find a subset $Λ_{ω,\{n_k\}}\subseteq \mathbb{R}$ such that $\big\{ e_λ(x) = e^{2πi λx}: λ\in Λ_{ω,\{n_k\}} \big\}$ forms an orthonormal basis for $L^2(μ_{ω,\{ n_k\}})$.

Theorems & Definitions (28)

• Theorem 1.1
• Remark
• Corollary 1.2
• Definition 1.3
• Theorem 1.4
• Example 1.5
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• proof