Intrinsic Diophantine approximation on quadric hypersurfaces

TL;DR

This paper investigates how well points on a quadric hypersurface can be approximated by rational points lying on the same hypersurface, revealing the influence of the quadratic form's ranks on approximation quality.

Contribution

It provides a comprehensive analysis of intrinsic Diophantine approximation on quadrics, linking it to the dynamics of projective transformation groups and exploring the roles of real and rational ranks.

Findings

Complete answers to key Diophantine approximation questions on quadrics.

Established a correspondence between approximation theory and group dynamics.

Identified the impact of quadratic form ranks on approximation properties.

Abstract

We consider the question of how well points in a quadric hypersurface $M\subset\mathbb R^d$ can be approximated by rational points of $\mathbb Q^d\cap M$. This contrasts with the more common setup of approximating points in a manifold by all rational points in $\mathbb Q^d$. We provide complete answers to major questions of Diophantine approximation in this context. Of particular interest are the impact of the real and rational ranks of the defining quadratic form, quantities whose roles in Diophantine approximation have never been previously elucidated. Our methods include a correspondence between the intrinsic Diophantine approximation theory on a rational quadric hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface, similar to earlier results in the non-intrinsic setting due to Dani ('86) and Kleinbock--Margulis ('99).