Periodicity of the spectrum in dimension one

Iosevich, Alex; Kolountzakis, Mihal N.

doi:10.2140/apde.2013.6.819

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Periodicity of the spectrum in dimension one

Alex Iosevich and Mihal N. Kolountzakis

Vol. 6 (2013), No. 4, 819–827

DOI: 10.2140/apde.2013.6.819

Abstract

A bounded measurable set $Ω$ , of Lebesgue measure 1, in the real line is called spectral if there is a set $Λ$ of real numbers (“frequencies”) such that the exponential functions $e_{λ} (x) = exp (2 π i λ x)$ , $λ \in Λ$ , form a complete orthonormal system of $L^{2} (Ω)$ . Such a set $Λ$ is called a spectrum of $Ω$ . In this note we prove that any spectrum $Λ$ of a bounded measurable set $Ω \subseteq ℝ$ must be periodic.

Keywords

spectrum, exponential bases, spectral sets, Fuglede's conjecture

Mathematical Subject Classification 2010

Primary: 42B05

Secondary: 42B99

Milestones

Received: 11 September 2011

Revised: 7 July 2012

Accepted: 1 September 2012

Published: 21 August 2013

Authors

Alex Iosevich
Department of Mathematics 915 Hylan Building University of Rochester Rochester, NY 14628 United States

Mihal N. Kolountzakis
Department of Mathematics University of Crete Knossos Avenue 71409 Iraklio Greece

Vol. 6, No. 4, 2013