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

For an expansive real matrix M=rho-1C0 rho-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,(alpha 1,alpha 2)t,(beta 1,beta 2)t}$D=\lbrace (0,0)<^>t,(\alpha _1,\alpha _2)<^>t,(\beta _1,\beta _2)<^>t\rbrace$ with alpha 2-2 beta 2 is not an element of 3Z$\alpha _2-2\beta _2\notin 3\mathbb {Z}$, let mu M,D$\mu _{M,D}$ be the self-affine measure defined by mu M,D(center dot)=13 n-ary sumation d is an element of D mu M,D(M(center dot)-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(mu M,D)$L<^>2(\mu _{M,D})$ contains an infinite orthogonal exponential set or mu M,D$\mu _{M,D}$ to be a spectral measure are given.