On the existence problem of regular Gabor frames

For every dimension $d>1$, we establish explicit criteria on lattices $\mathrm{\Lambda }\subset {\mathbb{R}}^{2d}$ with density $D(\mathrm{\Lambda })>1$ such that no function with a continuous Zak transform generates a Gabor frame along $\mathrm{\Lambda }$. 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.