Gabor orthonormal bases, tiling and periodicity

We show that if the Gabor system $\{g(x − t)e^{2\pi isx}, t\in T,s\in S\}$, is an orthonormal basis in $L^2(\mathbb{R})$ and if the window function $g$ is compactly supported, then both the time shift set $T$ and the frequency shift set $S$ must be periodic. To prove this we establish a necessary functional tiling type condition for Gabor orthonormal bases which may be of independent interest.publishe