Khinchin theorem for integral points on quadratic varieties

TL;DR

This paper extends the classical Khinchin theorem to integer points on quadratic varieties, using dynamical systems on homogeneous spaces to analyze approximation properties.

Contribution

It introduces a Khinchin-type result for quadratic varieties, linking Diophantine approximation with dynamics on orthogonal group homogeneous spaces.

Findings

Generic trajectories visit shrinking subsets infinitely often

Established a Diophantine approximation theorem for quadratic varieties

Connected dynamical systems with number-theoretic approximation

Abstract

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.