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

Abstract: 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 = \begin{bmatrix} \rho_1^{-1} & c 0 & \rho_2^{-1} \end{bmatrix} ) 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} \left( \frac{j \bm{a}}{p} + \mathbb{Z}^2 \right) ) for some ( \bm{a} \in \mathcal{E}{p}= { (i_1, i_2)^* : 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}{# 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 $\rho_1 \neq \rho_2$, with restricted digit sets $D$, we obtain a necessary and sufficient condition for $\mu_{M,D}$ to be a spectral measure.