By a Fourier quasicrystal we mean a pure point measure in R^d,
whose Fourier transform is also a pure point measure. This notion
was inspired by the experimental discovery of quasicrystalline
materials in the middle of 80's.
The classical example of such a measure comes from Poisson's
summation formula. Which other measures of this type may exist?
I will give the relevant background on this problem and present
our recent results obtained in joint work with Alexander Olevskii.