The dual spectral set conjecture

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 $\Omega'.$