An Amalgam Balian-Low Theorem for symplectic lattices of rational density

A Gabor space is a space generated by a discrete set of time-frequency shifted copies of a single window function. Starting from the question of whether a Gabor space contains additional time-frequency shifts of the window function we establish a new Balian-Low type result. This result extends (for example) the well established Amalgam Balian-Low Theorem in the one dimensional case. The Gabor spaces considered in this note are generated by symplectic lattices of rational density¹.