On the universal tiling conjecture in dimension one
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}\item[(i)] The Universal Tiling Conjecture is true for all \(p\in \mathbb{N}\); \item[(ii)] Every bounded Lebesgue measurable spectral set tiles by translations; \item[(iii)] For every \(\Omega=\bigcup_{i=1}^{n}(\alpha_{i}, \beta_{i})\) with \(\alpha_{i}, \beta_{i}\) rational numbers and \(|\Omega|=1\), if \(\Omega\) has a spectrum \(\Lambda\) with period \(p\), then \(\Omega\) tiles \(\mathbb{R}\) by a subset \(T\) of \(\frac{1}{p}\mathbb{Z}\); \item[(iv)] For every \(\Omega=\bigcup_{i=1}^{n}(\alpha_{i}, \beta_{i})\) with \(\alpha_{i}, \beta_{i}\in \mathbb{Z}\) and \(|\Omega|=N\), if \(\Omega\) has a spectrum \(\Lambda\) with minimal period \(\frac{r}{N}\), \(r\in \mathbb{Z}\), then \(\frac{N}{r}\) is an integer and \(\Omega\) tiles \(\mathbb{R}\) by a subset \(T\) of \(\frac{N}{r}\mathbb{Z}\). \end{itemize}}
scientific article