On the universal tiling conjecture in dimension one
type: Item
instance of: scholarly article
title: On the universal tiling conjecture in dimension one
zbMATH Open document ID: 1308.42003
author: Dorin Ervin Dutkay
author: Palle Jorgensen
published in: The Journal of Fourier Analysis and Applications
publication date: 2015-01-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1209.0666
review text: In this paper the authors prove some equivalences concerning the Fuglede conjecture and the Universal Tiling Conjecture. As preliminaries the following concepts and conjectures are considered . Definition. A Lebesgue measurable subset \(\Omega\subset \mathbb{R}\) with finite Lebesgue measure is called \textit{spectral}, if there exists a subset \(\Lambda\subset \mathbb{R}\) called \textit{spectrum for} \(\Omega\), such that \(\{e^{2\pi i\lambda x}; \lambda\in \Lambda\}\) is an orthogonal basis for \(L^{2}(\Omega)\). \(\Omega\) is said to tile \(\mathbb{R}\) by translations if there exists a subset \(T\subset \mathbb{R}\) such that \(\{\Omega+t; t\in T\}\) is a partition of \(\mathbb{R}\) up to Lebesgue measure zero. If \(\Omega\) is finite, then \(\Omega\) is called spectral if there exists a finite set \(\Lambda\), called spectrum for \(\Omega\), such that \(\{e^{2\pi i\lambda x}; \lambda\in \Lambda\}\) is a Hilbert space for \(L^{2}(\delta_{\Omega})\), where \(\delta_{\Omega}\) is the atomic measure \(\delta_{\Omega}=\sum_{a\in \Omega}\delta_{a}\) and \(\delta_{a}\) is the Dirac measure at \(a\). The following two conjectures are well-known: \textbf{Fuglede's conjecture.} \(\Omega\subset \mathbb{R}\) of finite Lebesgue measure is spectral, if and only if it tiles \(\mathbb{R}\) by translations. \textbf{Universal Tiling Conjecture.} Let \(p\in \mathbb{N}\) and \(\Gamma=\{0, \lambda_{1}, \dots, \lambda_{p-1}\}\) be a subset of \(\mathbb{R}\) with \(p\) elements, where \(\Gamma\) has the spectrum of the form \(\frac{1}{p}A\) with \(A\subset \mathbb{Z}\). Then, for every finite family \(A_{1}, \dots, A_{n}\subset \mathbb{Z}\) such that \(\frac{1}{p}A_{i}\) is a spectrum for \(\Gamma\) for all \(i\), there exists a tiling subset \(T\subset \mathbb{Z}\) such that \(A_{i}\) tiles \(\mathbb{Z}\) by \(T\) for all \(i\in \{1, \dots, n\}\). The main results of the paper can be stated as follows. Theorem. The following assertions are equivalent: {\parindent=8mm \begin{itemize…
Mathematics Subject Classification ID: 42A32
Mathematics Subject Classification ID: 05B45
zbMATH DE Number: 6384923
zbMATH Keywords: Fuglede conjecture
zbMATH Keywords: spectrum
zbMATH Keywords: universal tiling conjecture
zbMATH Keywords: Fourier analysis
reviewed by: Sorin G. Gal
MaRDI profile type: Publication
OpenAlex ID: W2104862866
Wikidata QID: Q123349983
arXiv ID: 1209.0666
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DOI: 10.1007/S00041-013-9264-7
description: scientific article