The Exact Number of Orthogonal Exponentials of a Class of Moran Measures on $\mathbb {R}^{3}$

Abstract

In this paper, we consider the nonspectralities of the Moran measure $\mu _{M,\{D_{n}\}}$ corresponding to an expanding integer matrix $M= {diag} [p,q,r]$ and the digit set

$$\begin{aligned} D_{n}=\left\{ {{\left( {\begin{array}{*{20}{c}} 0,0,0 \end{array}}\right) }^{T}}, {{\left( {\begin{array}{*{20}{c}} a_{n},0,0\\ \end{array}}\right) }^{T}},{{\left( {\begin{array}{*{20}{c}} 0,b_{n},0\\ \end{array}}\right) }^{T}},{{\left( {\begin{array}{*{20}{c}} 0,0,c_{n}\\ \end{array}}\right) }^{T}}\right\} ,\end{aligned}$$

where $p\in 2\mathbb {Z} \backslash \{0\}$, $q,r \in (2\mathbb {Z}+1)\backslash \{1,-1\}$, $a_{n},b_{n},c_{n}\in 2\mathbb {Z}+1$ and $\frac{b_{n}}{c_{n}}=\frac{b_{m}}{c_{m}}$. If $q=r$, we prove that there exists at most 4 mutually orthogonal exponential functions in the Hilbert space $L^{2}(\mu _{M,\{D_{n}\}})$, where the number 4 is the best upper bound. If $q=-r$, then there exists at most 8 mutually orthogonal exponential functions in the Hilbert space $L^{2}(\mu _{M,\{D_{n}\}})$, where the number 8 is the best upper bound. If $|q|\ne |r|$, then there is any number of orthogonal exponentials in $L^{2}(\mu _{M,\{D_{n}\}})$.