Spectral properties of a class of Sierpinski-type Moran measures on ℝⁿ

2025/05/14 by Ji‐Long Chen, Chen, Jia Long, Xiaoyu Yan +1
Mathematics · #28A80 #42C05 #46C05 #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2505.09360

openalex publication_date 2025/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let the infinite convolutions μ_\R_k\,\D_k\=δ_R₁^-1D₁*δ_R₁^-1R₂^-1D₂*δ_R₁^-1R₂^-1R₃^-1D₃*\dotsi be generated by the sequence of pairs \(\ (R_k,D_k) \_k=1^∞ \), where R_k∈ M_n(ℤ) is an expanding integer matric, D_k is a finite integer digit sets that satisfies the following two conditions: (i). \( # D_k = m \) and \( m>2 \) is a prime; (ii). \( \x: ∑_d∈ D_ke^{2πi⟨ d,x ⟩}=0\ =∪_i=1^ϕ(k)∪_j=1^m-1((j)/(m)ν_k,i+ℤⁿ) \) for some \( ν_k,i ∈ \ (l₁, ⋯, l_n)^t : l_i ∈ [1, m-1] ∩ ℤ, 1≤ i≤ n \ \). In this paper, we study the spectrality of μ_\R_k\,\D_k\, and some necessary and sufficient conditions for \( L²(μ_\R_k\,\D_k\) \) to have an orthogonal exponential function basis are established. Finally, we discuss the explanations and applications of our results.

Citations

• The Fuglede conjecture for convex domains is true in all dimensions
• Non-spectral problem for the planar self-affine measures
• Hadamard triples generate self-affine spectral measures
• Tiles with no spectra
• Tiles with no spectra
• Fuglede's conjecture is false in 5 and higher dimensions
• On spectral Cantor measures
• Fractal Geometry: Mathematical Foundations and Applications.

Related

• Non-spectrality of Moran measures with consecutive digits
• On the spectrality of a class of Moran measures
• Spectrality of Moran measures with finite arithmetic digit sets
• Spectrality of Moran-type measures with staggered contraction ratios
• Spectrality and non-spectrality of a class of Moran measures with three-element digits