Intrinsic Diophantine approximation on quadric hypersurfaces

We consider the question of how well points in a quadric hypersurface ${M\\subseteq\\mathbb{R}^d}$ can be approximated by rational ${\\mathbb{Q}^d\\cap M}$. This contrasts with more common setup approximating manifold all ${\\mathbb{Q}^d}$. provide complete answers to major questions Diophantine approximation this context. Of particular interest are impact real and ranks defining quadratic form, quantities whose roles have never been previously elucidated. Our methods include correspondence between intrinsic theory on dynamics group projective transformations which preserve that hypersurface, similar earlier results non-intrinsic setting due Dani (1986) Kleinbock–Margulis (1999).