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

2025/08/28 by Chen, Jia-Long, Ai, Wen-Hui
#28A80 #42C05 #46C05 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2508.20809

Abstract

We investigate the spectral properties of a class of Sierpinski-type self-affine measures defined by μ_M,D(⋅) = p^-1 ∑_{d ∈ D} μ_M,D(M(⋅) - d), where \( p \) is a prime number, \( M = \beginbmatrix ρ₁^-1 & c 0 & ρ₂^-1 \endbmatrix \) is a real upper triangular expanding matrix, and \( D = \d₀, d₁, ⋯, d_p-1\ ⊂ ℤ² \) satisfying \( Z(\widehatδ_D) = ∪_j=1^p-1 ( \fracj \bmap + ℤ² ) \) for some \( \bma ∈ E_p= \ (i₁, i₂)^* : i₁, i₂ ∈ [1, p-1] ∩ ℤ \ \), where \( Z(\widehatδ_D) \) denotes the set of zeros of \( \widehatδ_D \) with \( δ_D = (1)/(# D) ∑_{d ∈ D} δ_d \). When ρ₁ = ρ₂, we derive necessary and sufficient conditions for μ_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 L²(μ_M,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 μ_M,D to be a spectral measure.

Citations

• Hadamard triples generate self-affine spectral measures
• Complex Hadamard matrices and the Spectral Set Conjecture
• Tiles with no spectra
• Tiles with no spectra
• Fuglede's conjecture is false in 5 and higher dimensions

Related

• Spectrality of generalized Sierpinski-type self-affine measures
• Non-spectral problem for the planar self-affine measures
• Non-spectrality of self-affine measures on the three-dimensional Sierpinski gasket
• Spectra of the Sierpiński type spectral measure and their Beurling dimensions
• Fourier bases of a class of planar self-affine measures