Positive Fourier Quasicrystals, one-dimensional and higher dimensions

Lior Alon, MIT
Jadwin Hall A07

The concept of "quasi-periodic" sets, functions, and measures is prevalent in many fields including Mathematical Physics,

Fourier Analysis, and Number Theory. The Poisson summation formula provides a “Fourier characterization” for discrete periodic sets, saying that the Fourier transform of the counting measure of a discrete periodic set is also a counting measure of a discrete periodic set. Fourier Quasicrystals (FQ) generalize this notion of periodicity: a counting measure of a discrete set is called a Fourier quasicrystal (FQ) if its Fourier transform is also a discrete atomic measure, together with some growth condition.   

I will describe the recent progress on one-dimensional FQs: the Kurasov-Sarnak construction in terms of Lee-Yang (stable) polynomials, and our work on the distribution of gaps between atoms of such FQs, including examples with Poisson and CUE gap distributions.  

If time permits, I will describe our recent generalization of this construction to higher dimensions, and the physical questions that can now be asked. 

Based on joint works with Alex Cohen, Cynthia Vinzant, Mario Kummer, and Pavel Kurasov.