Integer tile and Spectrality of Cantor-Moran measures with equidifferent digit sets

2024/10/29 by Sha Wu, Wu, Sha, Yingqing Xiao +1
Mathematics · Materials Science · #Mathematical Dynamics and Fractals #Analytic and geometric function theory #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2410.21626

Abstract

Let \b_k\_k=1^∞ be a sequence of integers with |b_k|≥2 and \D_k\_k=1^∞ be a sequence of equidifferent digit sets with D_k=\0,1, ⋯, N-1\t_k, where N≥2 is a prime number and \t_k\_k=1^∞ is bounded. In this paper, we study the existence of the Cantor-Moran measure μ_\b_k\,\D_k\ and show that D_k:=D_k⊕ b_k D_k-1⊕ b_kb_k-1 D_k-2⊕⋯⊕ b_kb_k-1⋯ b₂D₁ is an integer tile for all k∈ℕ⁺ if and only if s_i\neqs_j for all i≠ j∈ℕ⁺, where s_i is defined as the numbers of factor N in (b₁b₂⋯ b_i)/(Nt_i). Moreover, we prove that D_k being an integer tile for all k∈ℕ⁺ is a necessary condition for the Cantor-Moran measure to be a spectral measure, and we provide an example to demonstrate that it cannot become a sufficient condition. Furthermore, under some additional assumptions, we establish that the Cantor-Moran measure to be a spectral measure is equivalent to D_k being an integer tile for all k∈ℕ⁺.

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