On the finite linear independence of lattice Gabor systems

Abstract: In the restricted setting of product phase space lattices, we give an alternate proof of P. Linnell's theorem on the finite linear independence of lattice Gabor systems in $L^2(\mathbb R^d)$. Our proof is based on a simple argument from the spectral theory of random Schr\"odinger operators; in the one-dimensional setting, we recover the full strength of Linnell's result for general lattices.