On the Spectral Properties of a Class of Planar Sierpinski Self-Affine Measures

We investigate the spectral properties of a class of Sierpinski-type self-affine measures defined by $${\mu }_{M,D}(\cdot )=p^{-1}\sum _{d\in D}{\mu }_{M,D}(M(\cdot )-d),$$ where $p$ is a prime number, $M=\left[\begin{array}{ccc}{\rho }_1^{-1}& c0& {\rho }_2^{-1}\end{array}\right]$ is a real upper triangular expanding matrix, and $D=\{d_0,d_1,\cdots ,d_{p-1}\}\subset {\mathbb{Z}}^2$ satisfying $\mathcal{Z}({\widehat{\delta }}_D)={\cup }_{j=1}^{p-1}(\frac{j\mathit{a}}{p}+{\mathbb{Z}}^2)$ for some $\mathit{a}\in {\mathcal{E}}_p=\{(i_1,i_2{)}^{\ast }:i_1,i_2\in [1,p-1]\cap \mathbb{Z}\}$, where $\mathcal{Z}({\widehat{\delta }}_D)$ denotes the set of zeros of ${\widehat{\delta }}_D$ with ${\delta }_D=\frac{1}{\mathrm{\#}D}{\sum }_{d\in D}{\delta }_d$. When ${\rho }_1={\rho }_2$, we derive necessary and sufficient conditions for ${\mu }_{M,D}$ to both: $(i)$ possess an infinite orthogonal set of exponential functions, and $(ii)$ be a spectral measure. When no infinite orthogonal exponential system exists in $L^2({\mu }_{M,D})$, we quantify the maximum number of orthogonal exponentials and provide precise estimates. For ${\rho }_1\mathrm{\ne }{\rho }_2$, with restricted digit sets $D$, we obtain a necessary and sufficient condition for ${\mu }_{M,D}$ to be a spectral measure.