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 \[ μM,D(·) = p⁻¹ ∑d ∈ D μM,D(M(·) - d), \] where \( p \) is a prime number, \( M = {bmatrix} ρ_1⁻¹ & c 0 & ρ_2⁻¹ {bmatrix} \) is a real upper triangular expanding matrix, and \( D = \{d_0, d_1, ⋯, dₚ₋₁\} ⊂ Z^2 \) satisfying \( Z(δD) = ∪ⱼ₌₁ᵖ⁻¹ ( {j {a}}{p} + Z^2 ) \) for some \( {a} ∈ Eₚ= \{ (i_1, i_2)^* : i_1, i_2 ∈ [1, p-1] ∩ Z \} \), where \( Z(δD) \) denotes the set of zeros of \( δD \) with \( δD = 1/# D ∑d ∈ D δ_d \). When ρ₁ = ρ₂, we derive necessary and sufficient conditions for μM,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 L²(μM,D), 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 μM,D to be a spectral measure.