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).
Highlights
We consider the question of how well points in a quadric hypersurface M ⊆ Rd can be approximated by rational points of Qd ∩ M
The hypersurface MQ0, which we study in detail in Section 9, has very interesting properties for intrinsic Diophantine approximation
We describe the relation between the intrinsic Diophantine approximation of a quadric hypersurface MQ satisfying pQ = pR = 1 and the approximation of points in the boundary of d-dimensional hyperbolic space Hd by parabolic fixed points in a lattice Γ ⊆ Isom(Hd) which depends on the quadric hypersurface MQ
Summary
Given a quadric hypersurface MQ ⊆ PdR satisfying pQ = pR = 1, there exists a lattice Γ ⊆ Isom(Hd) and a diffeomorphism Φ : ∂Hd → MQ such that if PΓ ⊆ ∂Hd is the set of parabolic fixed points of Γ, Φ(PΓ) = PdQ ∩ MQ This correspondence allows one to deduce the case pQ = pR = 1 of all the results of this subsection as consequences of known theorems about Diophantine approximation of lattices in Isom(Hd); see §3.4 for more detail.
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