Spectrality and non-spectrality of a class of Moran measures with three-element digits

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 $\{a_n,b_n\}=\{-1,1\}(\mathrm{mod}3)$. It is know that the infinite convolution of uniformly discrete probability measures $${\mu }_{\{p_n\},\{{\mathcal{D}}_n\}}:={\delta }_{p_1^{-1}\{0,a_1,b_1\}}\ast {\delta }_{(p_1{p}_2{)}^{-1}\{0,a_2,b_2\}}\ast \cdots$$ 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 ${\sup }_{n\ge 1}\{\frac{|a_n|+|b_n|}{|p_n|}\}<\infty$, we give two sufficient conditions under which that ${\mu }_{\{p_n\},\{{\mathcal{D}}_n\}}$ 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 ${\mu }_{p,\{{\mathcal{D}}_n\}}$ is not a spectral measure. Our results extend some known theorems in An et al. [JFA, 2019] and Lu et al. [JFAA, 2022].