Fourier Quasicrystal often refers to a discrete measure whose Fourier transform is also a discrete measure. In this talk, we present a work done by Nir Lev and Alexander Olevskii where they prove a conjecture raised by Jeffrey Lagarias: if both the supports of a measure µ and its Fourier transform are uniformly discrete sets, then the support of µ is contained in a finite union of translates of a certain lattice. Moreover, they show that under these assumptions, the measure μ is a finite linear combination of Dirac combs, translated and modulated. This is done in full generality for the one-dimensional case, and for higher dimensions under the assumption that the measure is non-negative.
https://univienna.zoom.us/j/66031419470?pwd=bXd3V0xEMWM0MTQwS09nWStEV0NnUT09