The dual spectral set conjecture
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Abstract:
Suppose that $\Lambda =(a\mathbf {Z}+b)\cup (c\mathbf {Z}+d)$ where $a,b,c,d$ are real numbers such that $a\neq 0$ and $c\neq 0.$ 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 $\Omega ’.$References
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Additional Information
- Steen Pedersen
- Affiliation: Department of Mathematics, Wright State University, Dayton, Ohio 45435
- MR Author ID: 247731
- Email: steen@math.wright.edu
- Received by editor(s): April 15, 2003
- Published electronically: February 6, 2004
- Communicated by: David R. Larson
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2053982