Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces

Abstract

In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's classic '76 paper to more recent results of Hersonsky and Paulin ('02, '04, '07). Concrete examples of situations we consider which have not been considered before include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point of any element of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater generality than any prior work of which we are aware, our results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of Bishop and Jones ('97) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson-Sullivan measure unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.

Random walks on homogeneous spaces and Diophantine approximation on fractals

“…We denote the set of uniformly radial points of Λ by UR Λ . Uniformly radial points can also be thought of as "badly approximable with respect to the parabolic points of Λ"; see [16,Proposition 1.21]. In particular, ‚ If H 2 is the upper half-plane model of hyperbolic geometry, then BH 2 " R, and the parabolic points of the lattice Λ def " PGL 2 pZq Ď G def " PGL 2 pRq are exactly the rational points of R (including 8).…”

“…In addition to these results, in what follows we will also prove several other Diophantine results about the measures supported on self-similar fractals, as well as considering analogous questions regarding intrinsic Diophantine approximation on spheres [29,15] and on Kleinian lattices (cf. [16] and the references therein).…”

Bounded orbits of diagonalizable flows on finite volume quotients of products of SL2(R)

“…Hyperplane potential game. The hyperplane potential game was introduced in [12] and also defines a class of subsets of R d called hyperplane potential winning (HPW for short) sets. The following lemma allows one to prove the HAW property of a set S ⊂ R d by showing that it is winning for the hyperplane potential game.…”

A Note on Badly Approximable Linear Forms on Manifolds

“…One of the reasons is that we apply a lemma from [21] that was formulated for classical Schmidt's games. However, it is clear that Theorems 2.1 and 2.4 will be true for the hyperplane absolute game (for the definition see [23]).…”