Related papers: Spectral sets and weak tiling

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $\Omega$ is a spectral set if and only if it can tile the…

Let $\Omega$ be a convex polytope in $\mathbb{R}^d$. We say that $\Omega$ is spectral if the space $L^2(\Omega)$ admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that…

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by…

A bounded set $\Omega \subset \mathbb{R}^d$ is called a spectral set if the space $L^2(\Omega)$ admits a complete orthogonal system of exponential functions. We prove that a cylindric set $\Omega$ is spectral if and only if its base is a…

The notion of weak tiling played a key role in the proof of Fuglede's spectral set conjecture for convex domains, due to the fact that every spectral set must weakly tile its complement. In this paper, we revisit the notion of weak tiling…

The notion of weak tiling was a key ingredient in the proof of Fuglede's spectral set conjecture for convex bodies \cite{conv}, due to the fact that every spectral set tiles its complement weakly with a suitable Borel measure. In this paper…

A conjecture of Fuglede states that a bounded measurable set D, of measure 1, can tile space by translations if and only if the Hilbert space L^2(D) has an orthonormal basis consisting of exponentials exp(i 2 pi lambda x). If D has the…

A conjecture of Fuglede states that a bounded measurable set $\Omega$ in space, of measure 1, can tile space by translations if and only if the Hilbert space $L^2(\Omega)$ has an orthonormal basis consisting of exponentials. If $\Omega$ has…

Fuglede's conjecture states that a subset $\Omega\subseteq\mathbb{R}^{n}$ of positive and finite Lebesgue measure is a spectral set if and only if it tiles $\mathbb{R}^{n}$ by translation. The conjecture does not hold in both directions for…

Let $\Omega\subset \mathbb{R}^d$ be a set of finite measure. The periodic tiling conjecture suggests that if $\Omega$ tiles $\mathbb{R}^d$ by translations then it admits at least one periodic tiling. Fuglede's conjecture suggests that…

Let $\Omega \subset \mathbb{R}$ be a compact set with measure $1$. If there exists a subset $\Lambda \subset \mathbb{R}$ such that the set of exponential functions $E_{\Lambda}:=\{e_\lambda(x) = e^{2\pi i \lambda x}|_\Omega :\lambda \in…

We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T.…

We investigate tiling properties of spectra of measures, i.e., sets $\Lambda$ in $\br$ such that $\{e^{2\pi i \lambda x}: \lambda\in\Lambda\}$ forms an orthogonal basis in $L^2(\mu)$, where $\mu$ is some finite Borel measure on $\br$. Such…

Fuglede's spectral set conjecture states that a subset $\Omega$ of a locally compact abelian group $G$ tiles the group by translation if and only if there exists a subset of continuous group characters which is an orthogonal basis of…

In this paper we study subsets $E$ of ${\Bbb Z}_p^d$ such that any function $f: E \to {\Bbb C}$ can be written as a linear combination of characters orthogonal with respect to $E$. We shall refer to such sets as spectral. In this context,…

We consider two number-theoretic problems arising from Fuglede's spectral set conjecture: characterizing finite sets that tile integers, and finding polynomials with (0,1) coefficients whose roots have a certain multiplicative structure. We…

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if $L^2(\Omega)$ admits an orthogonal basis of exponentials. While the product of spectral sets is known to be spectral, the converse fails in general. In this paper, we prove that…

The spectral set conjecture, also known as the Fuglede conjecture, asserts that every bounded spectral set is a tile and vice versa. While this conjecture remains open on ${\mathbb R}^1$, there are many results in the literature that…

We call a set $K \subset {\mathbb R}^s$ with positive Lebesgue measure a {\it spectral set} if $L^2(K)$ admits an exponential orthonormal basis. It was conjectured that $K$ is a spectral set if and only if $K$ is a tile (Fuglede's…

The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture in one dimension, which states that a bounded measurable subset of $\mathbb{R}$…