Khinchin's theorem for approximation by integral points on quadratic varieties

Abstract

We prove an analogue of the Khinchin's 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 of a certain geodesic flow visit a family of shrinking subsets infinitely often.