Spectrality of generalized Sierpinski-type self-affine measures

For an expanding integer matrix $M\in M_2(\mathbb{Z})$ and an integer digit set $D=\{(0,0)^t,(\alpha_1,\alpha_2)^t,(\beta_1,\beta_2)^t\}$ with $\alpha_1\beta_2-\alpha_2\beta_1\neq0$, let $\mu_{M,D}$ be the Sierpinski-type self-affine measure defined by $\mu_{M,D}(\cdot)=\frac{1}{3}\sum_{d\in D}\mu_{M,D}(M(\cdot)-d)$. In [5.36], the authors separately investigated the spectral property of the measure $\mu_{M,D}$ in the case of $\det(M)\notin 3\mathbb{Z}$ or $\alpha_1\beta_2-\alpha_2\beta_1\notin 3\mathbb{Z}$. In this paper, we consider the remaining case where $\det(M)\in 3\mathbb{Z}$ and $\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb{Z}$, and give the necessary and sufficient conditions for $\mu_{M,D}$ to be a spectral measure. This completely settles the spectrality of the Sierpinski-type self-affine measure $\mu_{M,D}$.