Fourier quasicrystals with positive integer masses are sections of Lee-Yang cycles

Source abstract

We characterize Fourier quasicrystal measures with positive integer masses as real sections of effective Lee-Yang cycles, with weighted local intersection degrees. The section matrix has positive maximal minors, and the mass at each root is its weighted local intersection degree. For Delone support the reduced support of the cycle can be chosen strict. The support of every integer-mass Fourier quasicrystal measure is the common complex zero set of finitely many real-valued trigonometric polynomials. For unit masses the representing cycle can be chosen reduced, with local degree one at every physical root. These results answer Questions (1), (2), and (4) of Alon, Kummer, Kurasov, and Vinzant. The proof recovers a finite-dimensional torus from the spectrum, proves that the resulting analytic set is algebraic, and reconstructs its complex branches and their integer coefficients from their moments. A support-height theorem for holonomic sequences gives the required linear spectral bound. Singular Lee-Yang sections are Fourier quasicrystal measures with their natural local intersection degrees; discarding those degrees can destroy the Fourier quasicrystal property.