Intrinsic Diophantine approximation on quadric hypersurfaces

Abstract

We consider the question of how well points in a quadric hypersurface $M\subseteq {\mathbb{R}}^d$ 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 (1986) and Kleinbock–Margulis (1999).