Spectrality and non-spectrality of a class of Moran measures with three-element digits
Xiao-Yu Yan, Wen-Hui Ai
TL;DR
This work tackles the spectrality of Moran measures with three-element digits on $\mathbb{R}$ by analyzing the infinite convolution $\mu_{\{p_n\},\{\mathcal{D}_n\}}$ with $\mathcal{D}_n=\{0,a_n,b_n\}$ and $p_n\in 3\mathbb{Z}\setminus\{0\}$. It derives two spectrality criteria that do not require the previously imposed bound $\sup_n\{(|a_n|+|b_n|)/|p_n|\}<\infty$: Theorem $\ref{thm1.2}$ uses a growth-controlled setup to construct a spectrum $\Lambda$ via $\lambda=(1/3)\sum_{k=1}^n d_k P_k$ with $d_k\in\{-1,0,1\}$, and Theorem $\ref{thm1.3}$ employs equi-positivity and Hadamard-triple techniques to obtain spectrality under weaker asymptotics, including a concrete example with $p_n=3n^2$, $a_n=3n^3+1$, $b_n=3n^3+2$. Additionally, the paper provides a non-spectrality criterion (Theorem $\ref{thm1.5}$) based on a maximal-mapping construction, showing non-spectrality can occur for constant $p$ under a subsequence relation $c\cdot q^{\omega_n}\in\{a_{\omega_n},b_{\omega_n}\}$ and giving explicit Moran non-spectral instances. Overall, the results extend prior work by removing the boundedness hypothesis and clarifying the balance between digit growth, tail Fourier decay, and spectral completeness, using Hadamard-triple concepts, equi-positivity, and maximal-mapping methods.
Abstract
A Borel probability measure \( μ\) with compact support on \( \mathbb{R}^n \) is called spectral measure if there exists a discrete set \( Λ\subset \mathbb{R}^n \) such that \( E_Λ:= \{e^{2πi \langle λ, x \rangle}: λ\in Λ\} \) forms an orthonormal basis of \( L^2(μ) \). In this paper, we study the spectrality and non-spectrality of a class of Moran measures with three-element digits on \( \mathbb{R} \). Let $p_n\in 3\mathbb Z\setminus\{0\}$ and $\mathcal{D}_n=\{0,a_n,b_n\}$ with $\{a_n,b_n\}=\{-1,1\}\pmod 3$. It is know that the infinite convolution of uniformly discrete probability measures $$μ_{\{p_n\},\{\mathcal D_n\}}:=δ_{p_1^{-1}\{0,a_1,b_1\}}\astδ_{(p_1p_2)^{-1}\{0,a_2,b_2\}}\ast\cdots $$ is a Moran measure with compact support if and only if \begin{align*} \sum_{n=1}^{\infty}|p_{1}p_{2}\cdots p_n|^{-1}d_n<\infty,\quad \mbox{where}\;d_n=\max\{0,|a_n|, |b_n|\}. \end{align*} Without the condition $\sup_{n\geq 1}\{\frac{|a_n|+|b_n|}{|p_n|}\}<\infty$, we give two sufficient conditions under which that $μ_{\{p_n\},\{\mathcal D_n\}}$ is a spectral measure. If $p_n=p>2$ and $\mathcal{D}_n=\{0,a_n,b_n\}$ with $\gcd(a_n,b_n)=1$, we also find an useful condition to guarantee that $μ_{p,\{\mathcal D_n\}}$ is not a spectral measure. Our results extend some known theorems in An et al. [JFA, 2019] and Lu et al. [JFAA, 2022].