The　well　known　density　theorem　in　time-frequency　analysis　establishes　the　connection　between　the　existence　of　a　Gabor　frame　G(g,　A,　B)　=　{e(2　pi　i　＜　Bm,x　＞)　g(x　-　An):　m,　n　is　an　element　of　Z(d)}　for　L-2(R-d)　and　the　density　of　the　time-frequency　lattice　AZ(d)　x　BZ(d).　This　is　also　tightly　related　to　lattice　tiling　and　packing.　In　this　paper　we　investigate　the　density　theorem　for　Gabor　systems　in　L-2(S)　with　S　being　an　AZ(d)-periodic　subset　of　R-d.　We　characterize　the　existence　of　a　Gabor　frame　for　L-2(S)　in　terms　of　a　condition　that　involves　the　Haar　measure　of　the　group　generated　by　AZ(d)　and　(B-t)(-1)Z(d).　This　new　characterization　is　used　to　recover　the　density　theorem　and　several　related　known　results　in　the　literature.　Additionally　we　apply　this　approach　to　obtain　the　density　theorems　for　multi-windowed　and　super　Gabor　frames　for　L-2(S).　(C)　2013　Elsevier　Inc.　All　rights　reserved.