A quantitative Balian-Low theorem for subspaces

We consider Gabor Riesz sequences generated by a window function with finite uncertainty product over a rational lattice in${\mathbb{R}^2}$. We prove that the distance of a time-frequency shift of the window function to the Gabor space is equivalent, up to constants, to the Euclidean distance of the parameters of the time-frequency shift to the lattice. Under certain additional assumptions, these constants can be estimated. As a byproduct of the methods employed, we also obtain a strengthening of the so-called weak Balian-Low theorem.