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

Li-Xiang An, Qian Li and Min-Min Zhang

Vol. 334 (2025), No. 2, 189–209

DOI: 10.2140/pjm.2025.334.189

Abstract

Suppose $b={\{b_n\}}_{n=1}^{\infty }$ is a sequence of integers bigger than 1 and $D={\{{\mathcal{𝒟}}_n\}}_{n=1}^{\infty }$ is a sequence of consecutive digit sets. Let ${\mu }_{b,D}$ be the Cantor–Moran measure defined by

$${\mu }_{b,D}={\delta }_{\frac{1}{b_1}{\mathcal{𝒟}}_1}\ast {\delta }_{\frac{1}{b_1{b}_2}{\mathcal{𝒟}}_2}\ast {\delta }_{\frac{1}{b_1{b}_2{b}_3}{\mathcal{𝒟}}_3}\ast \cdots .$$

We first prove that $L^2({\mu }_{b,D})$ possesses an exponential orthonormal basis if and only if $N_n$ divides $b_n$ for each $n\ge 2$. Subsequently, we show that the generalized Fuglede’s conjecture holds for such Cantor–Moran measures. An immediate consequence of this result is the equivalence between the existence of an exponential orthonormal basis and the integer-tiling of $D_n={\mathcal{𝒟}}_n+b_n{\mathcal{𝒟}}_{n-1}+b_2\cdots b_n{\mathcal{𝒟}}_1$ for $n\ge 1$.

Keywords

generalized Fuglede's conjecture, spectral measure, Cantor–Moran measure, orthogonal basis

Mathematical Subject Classification

Primary: 42C05, 46C05

Secondary: 28A25, 28A80

Milestones

Received: 29 September 2024

Revised: 1 January 2025

Accepted: 2 January 2025

Published: 8 February 2025