Rational approximation on quadrics: a simplex lemma and its consequences

TL;DR

This paper provides elementary proofs of enhanced results in intrinsic Diophantine approximation on rational quadric hypersurfaces, utilizing a refined simplex lemma to relate rational points and isotropic subspaces.

Contribution

It introduces a simplified proof of stronger Diophantine approximation results on quadrics through a new refinement of the simplex lemma.

Findings

Stronger versions of Diophantine approximation results on quadrics.

A refined simplex lemma linking rational points to isotropic subspaces.

Elementary proof techniques for complex approximation theorems.

Abstract

We give elementary proof 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 $X$.