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

TL;DR

This paper develops new criteria for Gaussian Gabor frames in multiple dimensions using advanced geometric methods, providing the first covolume-based condition applicable to most lattices and explicit bounds for frame stability.

Contribution

It introduces a novel covolume criterion for multivariate Gaussian Gabor frames based on Seshadri constants and Buser-Sarnak invariants, connecting frame theory with complex geometry.

Findings

Established sufficient conditions for frame sets using Seshadri constants.

Derived explicit estimates for frame bounds via Robin constants.

Provided a sharp Robin constant estimate in one dimension using Arakelov geometry.

Abstract

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. The recent Berndtsson--Lempert method and the Ohsawa--Takegoshi extension theorem also allow us to give explicit estimates for the frame bounds in terms of certain Robin constant. In the one-dimensional case we obtain a sharp estimate of the Robin constant using Faltings' theta metric formula for the Arakelov Green functions.