Vol. 6, No. 4, 2013

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

Keywords
spectrum, exponential bases, spectral sets, Fuglede's conjecture
Primary: 42B05
Secondary: 42B99