The Spectrality of a Class of Fractal Measures on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R}^{n}$$\end{document}

Let M=ρ−1I∈Mn(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=\rho^{-1}I\in M_{n}(\mathbb{R})$$\end{document} be an expanding matrix with 0 < ∣ ρ ∣ < 1 and D⊂Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\subset\mathbb{Z}^{n}$$\end{document} be a finite digit set with 0 ∈ D and Z(mD)−Z(mD)⊂Z(mD)∪{0}⊂m−1Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{Z}(m_{D})-\cal{Z}(m_{D})\subset\cal{Z}(m_{D})\cup\{0\}\subset m^{-1}\mathbb{Z}^{n}$$\end{document} for a prime m, where Z(mD):={x:∑d∈De2πi⟨λ,x⟩=0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{Z}(m_{D}):=\{x : \sum\nolimits_{d\in D}e^{2\pi \mathrm{i}\langle\lambda,x\rangle}=0\}$$\end{document}. Let μM,D be the associated self-similar measure defined by μM,D(⋅)=1|D|∑d∈DμM,D(M(⋅)−d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu_{M,D}(\cdot)={1\over{\vert D\vert}}\sum\nolimits_{d\in D}\mu_{M,D}(M(\cdot)-d)$$\end{document}. In this paper, the necessary and sufficient conditions for L2(μM,D) to admit infinite orthogonal exponential functions are given. Moreover, by using the order theory of polynomial, we estimate the number of orthogonal exponential functions for all cases that L2(μM,D) does not admit infinite orthogonal exponential functions.