Jorgensen–Pedersen dual spectral set conjecture

Let $\Gamma\subset\mathbb R$ be a set. A spectrum for a spectral set $\Omega$ is a set of frequencies whose normalized exponentials form an orthonormal basis of $L^2(\Omega)$. A prototile $T$ is a measurable set that tiles $\mathbb R$ by translations, and a tiling set for $T$ is a set of translation vectors whose translates partition $\mathbb R$ almost everywhere. The dual spectral set conjecture. A subset $\Gamma$ of $\mathbb R$ is a spectrum for some spectral set $\Omega$ if and only if it is a tiling set for some prototile $T$. This is the dual formulation of the spectral-set/tiling correspondence, relating frequency sets that occur as spectra to translation sets that occur in tilings. The source presents it as open in the surrounding discussion.