Rational approximation on quadrics: a simplex lemma and its consequences

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 $X$.