Density results for Gabor systems associated with periodic subsets of the real line

The well-known density theorem for one-dimensional Gabor systems of the form (e2Jl'imbx g(x-na)}m,nEZ, where g E L2(JR), states that a necessary and sufficient condition for the existence of such a systemwhQse linear span is dense in L2(R), or which forms a frame for L2(R), is that the density condition a b :s I is satisfied. The main goal of this paper is to study the analogous problem for Gabor systems for which the window function g vanishes outside a periodic set 5 C JR which is a Z-shift invariant. We obtain measure-theoretic conditions that are necessary and sufficient for the existence of a window g such that the linear span of the corresponding Gabor system is dense in L2(5). Moreover, we show that if this density condition holds, there exists, in fact, a measurable set E C JR with the property that the Gabor system associated with the same parameters a, b and the window g = XE, forms a tight frame for L2(5).