Vol. 6, No. 4, 2013

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Periodicity of the spectrum in dimension one

Alex Iosevich and Mihal N. Kolountzakis

Vol. 6 (2013), No. 4, 819–827
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 eλ(x) = exp(2πiλx), λ Λ, form a complete orthonormal system of L2(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 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