The dual spectral set conjecture

Pedersen, Steen

doi:10.1090/S0002-9939-04-07403-9

The dual spectral set conjecture
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Proc. Amer. Math. Soc. 132 (2004), 2095-2101 Request permission

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 $\Omega ’.$

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