The generalized Fuglede's conjecture holds for a class of Cantor-Moran measures

TL;DR

The paper characterizes when Cantor-Moran measures $μ_{\mathbf{b},\mathbf{D}}$ admit exponential orthonormal bases by establishing equivalence with a generalized Fuglede tiling identity: $μ_{\mathbf{b},\mathbf{D}} * ν = \mathcal{L}_{[0,N_1/b_1]}$. It proves that, for the class ${\mathbf{D}}_n = {\mathcal{D}}_n + b_n{\mathcal{D}}_{n-1} + \cdots + b_2\cdots b_n {\mathcal{D}}_1$ of iterated digits, spectrality is equivalent to each ${\mathbf{D}}_n$ being an integer tile, which in turn is equivalent to $N_n \mid b_n$ for all $n\ge 2$. The authors develop a framework based on weak convergence of convolutions, bi-zero sets, and suitable spectra decompositions to connect global spectrality with finite truncations $μ_n$, and use tiling arguments to bridge spectral measures and integer tiling. The results extend the scope of the generalized Fuglede conjecture to a broad family of Cantor-Moran measures and clarify when spectral and tiling phenomena coincide in fractal settings.

Abstract

Suppose ${\bf b}=\{b_n\}_{n=1}^{\infty}$ is a sequence of integers bigger than 1 and ${\bf D}=\{{\mathcal D}_{n}\}_{n=1}^{\infty}$ is a sequence of consecutive digit sets. Let $μ_{{\bf b},{\bf D}}$ be the Cantor-Moran measure defined by \begin{eqnarray*} μ_{{\bf b},{\bf D}}&=& δ_{\frac{1}{b_1}{\mathcal D}_{1}}\astδ_{\frac{1}{b_1b_2}{\mathcal D}_{2}}\ast δ_{\frac{1}{b_1b_2b_3}{\mathcal D}_{3}}\ast\cdots. \end{eqnarray*} We prove that $L^2(μ_{{\bf b},{\bf D}})$ possesses an exponential orthonormal basis if and only if $μ_{{\bf b},{\bf D}}\astν={\mathcal L}_{[0,N_1/b_1]}$ for some Borel probability measure $ν$. This theorem shows that the generalized Fuglede's conjecture is true for such Cantor-Moran measure. An immediate consequence of this result is the equivalence between the existence of an exponential orthonormal basis and the integral tiling of ${\bf D}_n={\mathcal D}_{n}+b_n{\mathcal D}_{n-1}+b_2\cdots b_n{\mathcal D}_{1}$ for $n\geq1$.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Corollary 2.2
• proof
• Theorem 2.3: JP1998LMW20222
• Theorem 2.4