A characterization of Gabor Riesz bases with separable time-frequency shifts

A Gabor system generated by a window function $g\in L^2(\mathbb{R}^d)$ and a separable set $\Lambda\times \Gamma \subset \mathbb{R}^{2d}$ is the collection of time-frequency shifts of $g$ given by $\mathcal G(g, \Lambda\times \Gamma) = \left\{ e^{2\pi i \xi\cdot t}g(t-x)\right\}_{ (x,\xi)\in \Lambda\times \Gamma }$. One of the fundamental problems in Gabor analysis is to characterize all windows and time-frequency sets that generate a Gabor frame or Gabor orthonormal basis. The case of Gabor orthonormal bases generated by characteristic functions $g=\chi_\Omega$ has been solved by Han and Wang. In this paper, we build on these results and obtain a full characterization of Riesz Gabor systems of the form $\mathcal G(\chi_\Omega, \Lambda\times \Gamma)$ when $\Omega$ is a tiling of $\mathbb{R}^d$ with respect to $\Lambda$. Furthermore, for a certain class of lattices $\Lambda\times \Gamma$, we prove that a necessary condition for the characteristic function of a multi-tiling set to serve as a window function for a Riesz Gabor basis is that the set must be a tiling set. To prove this, we develop new results on the zeros of the Zak transform and connect these results to Gabor frames.