We investigate the question of how well points on a nondegenerate k k -dimensional submanifold M ⊆ R d M \subseteq \mathbb R^d can be approximated by rationals also lying on M M , establishing an upper bound on the “intrinsic Dirichlet exponent” for M M . We show that relative to this exponent, the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure. Our bound on the intrinsic Dirichlet exponent is phrased in terms of an explicit function of k k and d d which does not seem to have appeared in the literature previously. It is shown to be optimal for several particular cases. The requirement that the rationals lie on M M distinguishes this question from the more common context of (ambient) Diophantine approximation on manifolds, and necessitates the development of new techniques. Our main tool is an analogue of the simplex lemma for rationals lying on M M which provides new insights on the local distribution of rational points on nondegenerate manifolds.

We quantify the density of rational points in the unit sphere S n , proving analogues of the classical theorems on the embedding of Q n into R n . Specifically, we prove a Dirichlet theorem stating that every point α ∈ S n is sufficiently approximable, the optimality of this approximation via the existence of badly approximable points, and a Khintchine theorem showing that the Lebesgue measure of approximable points is either zero or full depending on the convergence or divergence of a certain sum. These results complement and improve on previous results, particularly recent theorems of Ghosh, Gorodnik and Nevo.Date: May 22, 2013.

For a proper, geodesic, Gromov hyperbolic metric space X, a discrete subgroup of isometries Γ whose limit set is uniformly perfect, and a disjoint collection of horoballs {Hj}, we show that the set of limit points badly approximable by {Hj} is absolutely winning in the limit set Λ(Γ). As an application, we deduce that for a geometrically finite Kleinian group acting on H n+1 , the limit points badly approximable by parabolics, denoted BA(Γ), is absolutely winning, generalizing previous results of Dani and McMullen. As a consequence of winning, we show that BA(Γ) has dimension equal to the critical exponent of the group. Since BA(Γ) can alternatively be described as the limit points representing bounded geodesics in the quotient H n+1 /Γ, we recapture a result originally due to Bishop and Jones.

We consider the question of how well points in a quadric hypersurface M Â R d can be approximated by rational points of Q d \ M . This contrasts with the more common setup of approximating points in a manifold by all rational points in Q d . We provide complete answers to major questions of Diophantine approximation in this context. Of particular interest are the impact of the real and rational ranks of the defining quadratic form, quantities whose roles in Diophantine approximation have never been previously elucidated. Our methods include a correspondence between the intrinsic Diophantine approximation theory on a rational quadric hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface, similar to earlier results in the non-intrinsic setting due to Dani (1986) and Kleinbock-Margulis (1999).

We prove an analogue of a theorem of A. Pollington and S. Velani ('05), furnishing an upper bound on the Hausdorff dimension of certain subsets of the set of very well intrinsically approximable points on a quadratic hypersurface. The proof incorporates the framework of intrinsic approximation on such hypersurfaces first developed in the authors' joint work with D. Kleinbock (preprint '14) with ideas from work of D. Kleinbock, E. Lindenstrauss, and B. Weiss ('04).