Universal spectra, universal tiling sets and the spectral set conjecture

Authors

  • Steen Pedersen
  • Yang Wang

DOI:

https://doi.org/10.7146/math.scand.a-14325

Abstract

A subset $\Omega$ of $\mathbf{R}^d$ with finite positive Lebesgue measure is called a spectral set if there exists a subset $\Lambda\subset\mathbf{R}$ such that ${\mathcal E}_\Lambda :=\{e^{i2\pi \langle\lambda, x\rangle}: \lambda\in\Lambda\}$ form an orthogonal basis of $L^2(\Omega)$. The set $\Lambda$ is called a spectrum of the set $\Omega$. The Spectral Set Conjecture states that $\Omega$ is a spectral set if and only if $\Omega$ tiles $\mathbf{R}^d$ by translation. In this paper we prove the Spectral Set Conjecture for a class of sets $\Omega \subset \mathbf{R}$. Specifically we show that a spectral set possessing a spectrum that is a strongly periodic set must tile $\mathbf{R}$ by translates of a strongly periodic set depending only on the spectrum, and vice versa.

Downloads

Published

2001-06-01

How to Cite

Pedersen, S., & Wang, Y. (2001). Universal spectra, universal tiling sets and the spectral set conjecture. MATHEMATICA SCANDINAVICA, 88(2), 246–256. https://doi.org/10.7146/math.scand.a-14325

Issue

Section

Articles