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

Abstract

For a prime number $p>2$, let $0\in D\subset {\mathbb{Z}}^n$ be a $p$-element digit set satisfying $\mathcal{Z}({\hat{\delta }}_D)={\cup }_{j=1}^{p-1}(\frac{j}{p}\mathbf{a}+{\mathbb{Z}}^n)$ 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 $\mathcal{Z}({\hat{\delta }}_D)$ 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 \[ μ_{Q,D}(\cdot) = \frac{1}{\#D} \sum_{d \in D} μ_{Q,D}(Q(\cdot) - d). \] In this paper, we first provide sufficient condition for a maximal orthogonal family $E_{\Lambda }=\{e^{-2\pi i⟨\lambda ,x⟩}:\lambda \in \Lambda \subset {\mathbb{R}}^n\}$ to be an orthogonal basis of $L^2({\mu }_{Q,D})$ when $Q=p\mathrm{diag}[q,\dots ,q]$ with $q\ge 1$. 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 $Q=p\mathrm{diag}[l_1,\dots ,l_n]$ 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 ${\mu }_{Q,D}$.