Gabor frames with rational density

We consider the Gabor system G(g,α,β)={e2πiβntg(t−αm):m,n∈Z} for the case of rational oversampling, i.e. αβ∈Q and prove that for any odd window function g this system does not form a frame if , n=2,3,… . We also construct a ‘rational’ analogue of the Ron–Shen Gramian.