标题： 关于一类平面Sierpinski自仿测度的谱性质 显示英文标题

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

摘要： 我们研究由以下定义的Sierpinski型自仿测度的谱性质 \[ \mu_{M,D}(\cdot) = p^{-1} \sum_{d \in D} \mu_{M,D}(M(\cdot) - d), \] 其中 \( p \)是一个素数， \( M = \begin{bmatrix} \rho_1^{-1} & c 0 & \rho_2^{-1} \end{bmatrix} \)是一个实上三角扩张矩阵，且 \( D = \{d_0, d_1, \cdots, d_{p-1}\} \subset \mathbb{Z}^2 \)满足 \( \mathcal{Z}(\widehat{\delta}_{D}) = \cup_{j=1}^{p-1} \left( \frac{j \bm{a}}{p} + \mathbb{Z}^2 \right) \)对某个 \( \bm{a} \in \mathcal{E}_{p}= \{ (i_1, i_2)^* : i_1, i_2 \in [1, p-1] \cap \mathbb{Z} \} \)，其中 \( \mathcal{Z}(\widehat{\delta}_{D}) \)表示 \( \widehat{\delta}_{D} \)的零点集，其中 \( \delta_{D} = \frac{1}{\# D} \sum_{d \in D} \delta_d \)。 当$\rho_1 = \rho_2$时，我们推导出$\mu_{M,D}$的必要且充分条件，使其同时满足：$(i)$拥有无限的指数函数正交集，以及$(ii)$是一个谱测度。 当$L^{2}(\mu_{M,D})$中不存在无限的正交指数系统时，我们量化了正交指数的最大数量并提供精确的估计。 对于$\rho_1 \neq \rho_2$，在限制数字集$D$的情况下，我们得到了$\mu_{M,D}$成为谱测度的充要条件。 显示英文摘要

摘要： 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.