Simons Semester in Banach Center: "Dynamical Systems"

Alexander Olevskii: Fourier quasicrystals

Alexander Olevskii (Tel Aviv University)

Fourier quasicrystals


By a Fourier quasicrystal one often means a discrete measure in R^n with pure point spectrum. The classic example of such measure is given by the Poisson summation formula. Meyer's "model sets" provide many examples of aperiodic measures with uniformly discrete support and dense point spectrum. A new peak of interest to the subject appeared in the middle of 80th, after the experimental discovery of physical quasicrystals. It has been conjectured that a measure whose the support and the spectrum both are uniformly discrete has periodic structure. The conjecture was proved recently in our joint paper with N. Lev. Our approach is based on the interaction of some problems in harmonic analysis and discrete geometry. I am going to present necessary background and discuss this and related results.