We investigate the spectral properties of a class of Sierpinski-type self-affine measures defined by \ μ₌, ₃ () = p^-1 ₃ ₃ μ₌, ₃ (M () - d), \ where \ (p \) is a prime number, \ (M = bmatrix ρ₁^-1 & c 0 & ρ₂^-1 bmatrix \) is a real upper triangular expanding matrix, and \ (D = \d₀, d₁, , d-₁\ Z² \) satisfying \ (Z (δ₃) = ₉=₁^p-1 (j ap + Z²) \) for some \ (a E= \ (i₁, i₂) ^*: i₁, i₂ 1, p-1 Z \ \), where \ (Z (δ₃) \) denotes the set of zeros of \ (δ₃ \) with \ (δ₃ = 1\# D ₃ ₃ δd \). When ρ₁ = ρ₂, we derive necessary and sufficient conditions for μ₌, ₃ 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 (μ₌, ₃), we quantify the maximum number of orthogonal exponentials and provide precise estimates. For ρ₁ ρ₂, with restricted digit sets D, we obtain a necessary and sufficient condition for μ₌, ₃ to be a spectral measure.
Chen et al. (Thu,) studied this question.