Gabor frames for quasicrystals, K-theory, and twisted gap labeling

We study the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal Λ, and the K-theory of an associated twisted groupoid algebra. In particular, we construct a finitely generated projective module over this algebra, and any multiwindow Gabor frame for Λ can be used to construct an idempotent representing this module in K-theory. For lattice subsets in dimension two, this allows us to prove a twisted version of Bellissard's gap labeling theorem.