Khinchin theorem for integral points on quadratic varieties
  We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often. 
Alexander Gorodnik, Nimish A. Shah