Gaussian Gabor frames, Seshadri constants and generalized Buser--Sarnak invariants

Understand

We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from K\"ahler geometry such as H\"ormander's $\dbar$-$L^2$ estimate with singular weight, Demailly's Calabi--Yau method for K\"ahler currents and a K\"ahler-variant generalization of the symplectic embedding theorem of McDuff--Polterovich for ellipsoids.

• Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames.
• We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice.
• In particular, we obtain an effective Gaussian Gabor frame criterion in terms of the covolume for almost all lattices, which is the first general covolume criterion in multivariate Gaussian Gabor frame theory.