Gabor orthonormal bases, tiling and periodicity

We show that if the Gabor system { g ( x - t ) e 2 π i s x } , t ∈ T , s ∈ S , is an orthonormal basis in L 2 ( 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.