2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74683We 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 correctionsGroup TheoryNumber Theory11J83, 22E40, 53C35text