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The dual spectral set conjecture


Author: Steen Pedersen
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2095-2101
MSC (2000): Primary 42A99, 42C99, 51M04, 52C99
DOI: https://doi.org/10.1090/S0002-9939-04-07403-9
Published electronically: February 6, 2004
MathSciNet review: 2053982
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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $\Lambda=(a\mathbf{Z}+b)\cup(c\mathbf{Z}+d)$ where $a,b,c,d$ are real numbers such that $a\neq0$ and $c\neq0.$ The union is not assumed to be disjoint. It is shown that the translates $\Omega+\lambda$, $\lambda\in\Lambda$, tile the real line for some bounded measurable set $\Omega$ if and only if the exponentials $e_{\lambda}(x)=e^{i2\pi\lambda x}$, $\lambda\in\Lambda$, form an orthogonal basis for some bounded measurable set $\Omega'.$


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Additional Information

Steen Pedersen
Affiliation: Department of Mathematics, Wright State University, Dayton, Ohio 45435
Email: steen@math.wright.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07403-9
Keywords: Fourier basis, non-harmonic Fourier series, tiling, spectral set, spectral pair
Received by editor(s): April 15, 2003
Published electronically: February 6, 2004
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society

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