We particularly mention those collected and stated by Lagarias, some of which have recently been answered by Kellendonk and Lenz, by Favorov and by Lev and Olevskii. While the natural setting of tempered distributions simplifies the harmonic analysis in this case significantly, and powerful complex-analytic techniques may be applied, several interesting open problems remain. In particular, we will be able to classify, in Theorem 1, the few cases of Fourier-transformable measures that are supported on cut and project sets and have a sparse Fourier transform. Although model sets typically lead to diffraction measures with dense support, the methods from this field provide immensely useful tools for the questions at hand. Here, Meyer's pioneering work on model sets plays a key role see for a detailed account, and for an exposition of their appearance in diffraction theory.
The understanding of such measures, and translation-bounded measures and their transforms in general, has reached a reasonably mature state for G = R d, where they arise in the study of quasicrystals.
Δ Γ ̂ = dens ( Γ ) δ Γ 0, (1)where Γ 0 denotes the dual lattice of Γ see and references therein for background.