Spectral structure and eigenmatrices of self-affine measures with $p$-digits in $ \bf{\mathbb{R}^{n}} $

Abstract

For a prime number $p > 2$, let $\bm{0} \in D \subset \mathbb{Z}^n$ be a $p$-element digit set satisfying $ \mathcal{Z}(\widehat{\delta}_D) =\cup_{j=1}^{p-1}(\frac{j}{p}\bm{a}+\mathbb{Z}^{n}) $ for some \( \bm{a} \in \{ (i_1, \dots, i_n)^t : i_k \in [1, p-1] \cap \mathbb{Z}, 1\leq k\leq n \} \), where $\mathcal{Z}(\widehat{\delta}_D)$ is the zero set of the Fourier transform of $\delta_D$. Let $Q$ be an integer expansive diagonal matrix in $\mathbb{R}^{n}$, the self-affine measure $\mu_{Q,D}$ is defined by \[ \mu_{Q,D}(\cdot) = \frac{1}{\#D} \sum_{d \in D} \mu_{Q,D}(Q(\cdot) - d). \] In this paper, we first provide sufficient condition for a maximal orthogonal family $E_{\Lambda}=\{ e^{-2\pi i \langle \lambda, x \rangle} : \lambda\in \Lambda \subset\mathbb{R}^{n}\}$ to be an orthogonal basis of $L^2(\mu_{Q,D})$ when $Q = p\operatorname{diag}[q, \dots, q]$ with $q\geq 1$. Then we obtain necessary and sufficient conditions for the integer matrix $R$ such that $E_{\Lambda}$ and $E_{R\Lambda}$ are both orthogonal basis of $L^{2}(\mu_{Q,D})$. Furthermore, for $Q = p\operatorname{diag}[l_1, \dots, l_n]$ with $|l_1 \dots l_n| > 1$, we give a necessary and sufficient condition under which the real diagonal matrix is the second type spectral eigenmatrices of $\mu_{Q,D}$.