Rational approximation on quadrics: A simplex lemma and its consequences

Abstract

We give elementary proofs of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces $X\subset {\mathbb{P}}^n(\mathbb{R})$. The main tool is a refinement of the simplex lemma, which essentially says that rational points on $X$ which are sufficiently close to each other must lie on a totally isotropic rational subspace of $X$.