Spectrality of a class of Cantor-dust type self-affine tilesSpectrality of a class of Cantor-dust...J.-c. Liu, J.-j. Wang et al.

被引：0

作者：

Jing-cheng Liu ^{[1
]}

Jia-jie Wang ^{[1
]}

Jia Zheng ^{[2
]}

机构：

[1] Hunan Normal University,Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics

[2] Wuhan University,School of Mathematics and Statistics

来源：

Bulletin of the Malaysian Mathematical Sciences Society | 2025年 / 48卷 / 4期

关键词：

Spectral set; Self-affine set; Tile; Fuglede’s conjecture; Primary 28A80; Secondary 42C05; 46C05;

D O I：

10.1007/s40840-025-01882-w

中图分类号：

学科分类号：

摘要：

We say that a set Ω⊆Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subseteq {\mathbb {R}}^n$$\end{document} with positive Lebesgue measure is a spectral set if L2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}(\Omega )$$\end{document} admits an orthogonal basis of exponentials. In this paper, we study the spectrality of self-affine tiles. We prove a class of sets T:=T(M,D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T:=T(M,D)$$\end{document} with positive Lebesgue measure satisfying MT=⋃d∈D(T+d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$MT=\bigcup _{d\in D}(T+d)$$\end{document} are spectral sets, where M∈M2(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\in M_{2}({\mathbb {Z}})$$\end{document} is an expanding matrix with |det(M)|=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\det {(M)}|=4$$\end{document} and D={0,α,β,-(α+β)}⊂Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D=\{{\textbf{0}},\mathbf {\alpha },\beta ,-(\alpha +\beta )\}\subset {\mathbb {Z}}^2$$\end{document} is a non-collinear digit set. As an application, we conclude that T is a spectral set if and only if it tiles R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^2$$\end{document} by translations.

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