Fourier bases of a class of planar self-affine measures

Abstract

Let $\mu_{M,D}$ be the planar self-affine measure generated by an expansive integer matrix $M\in M_2(\mathbb{Z})$ and a non-collinear integer digit set $D=\left\{\begin{pmatrix} 0\\0\end{pmatrix},\begin{pmatrix} \alpha_{1}\\ \alpha_{2} \end{pmatrix}, \begin{pmatrix} \beta_{1}\\ \beta_{2} \end{pmatrix}, \begin{pmatrix} -\alpha_{1}-\beta_{1}\\ -\alpha_{2}-\beta_{2} \end{pmatrix}\right\}$. In this paper, we show that $\mu_{M,D}$ is a spectral measure if and only if there exists a matrix $Q\in M_2(\mathbb{R})$ such that $(\tilde{M},\tilde{D})$ is admissible, where $\tilde{M}=QMQ^{-1}$ and $\tilde{D}=QD$. In particular, when $\alpha_1\beta_2-\alpha_2\beta_1\notin 2\Bbb Z$, $\mu_{M,D}$ is a spectral measure if and only if $M\in M_2(2\mathbb{Z})$.