The concept of "quasiperiodic" sets, functions, and measures is prevalent in diverse mathematical fields such as Mathematical Physics, Fourier Analysis, and Number Theory. The Poisson summation formula provides a natural characterization of quasiperiodicity: a counting measure of a discrete set is a Fourier quasicrystal (FQ) if its Fourier transform is also a discrete atomic measure, together with some growth condition.

Recently, Kurasov and Sarnak provided a method for constructing one dimensional FQs as the return times of a linear flow along an irrational slope on a torus to the zero set of a multivariate Lee-Yang polynomial. In this talk, I will show that, in fact, every one dimensional FQ admits such a construction. I will also discuss the distribution of gaps in one dimensional FQs, showing that they are dense in an interval, and the distribution is given explicitly in terms of the slope and polynomial in the Kurasov-Sarnak construction. In the last part I will talk about a generalization of their construction to any dimension.

The talk is based on joint works with Alex Cohen, Cynthia Vinzant, Mario Kummer, and Pavel Kurasov.