On the existence problem of regular Gabor frames

Abstract

For every dimension $d > 1$, we establish explicit criteria on lattices $Λ\subset \mathbb{R}^{2d}$ with density $D(Λ) > 1$ such that no function with a continuous Zak transform generates a Gabor frame along $Λ$. In particular, this gives a negative answer to the existence problem of Gabor frames with window functions in the Schwartz space, the Feichtinger algebra, and the Fourier-invariant Wiener space. Our result is based on a characterization of when a collection of quasiperiodic functions admits a common zero, which may be of independent interest. We also include a formalization of our main result in Lean 4.