Fourier bases of the planar self‐affine measures with three digits

For an expansive real matrix M=ρ−1C0ρ−1$M= \def\eqcellsep{&}\begin{bmatrix} \rho ^{-1} & \mathcal {C}\\ 0& \rho ^{-1} \end{bmatrix}$ and a noncollinear integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t}$D=\lbrace (0,0)^t,(\alpha _1,\alpha _2)^t,(\beta _1,\beta _2)^t\rbrace$ with α2−2β2∉3Z$\alpha _2-2\beta _2\notin 3\mathbb {Z}$, let μM,D$\mu _{M,D}$ be the self‐affine measure defined by μM,D(·)=13∑d∈DμM,D(M(·)−d)$\mu _{M,D}(\cdot )=\frac{1}{3}\sum _{d\in D}\mu _{M,D}(M(\cdot )-d)$. In this paper, some necessary and sufficient conditions for L2(μM,D)$L^2(\mu _{M,D})$ contains an infinite orthogonal exponential set or μM,D$\mu _{M,D}$ to be a spectral measure are given.