Diophantine approximation on rational quadrics

Abstract

We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays with a fixed maximal singular direction, which move away into one end of a locally symmetric space at linear depth, infinitely many times.

55 pages, 3 figures, to appear in Math. Annalen, revised version : updated references, minor corrections