An application of metric diophantine approximation in hyperbolic space to quadratic forms

For any real $\tau$, a $\lim\sup$ set $W_{G,y}(\tau)$ of $\tau$-(well)-approximable points is defined for discrete groups $G$ acting on the Poincaré model of hyperbolic space. Here $y$ is a 'distinguished point' on the sphere at infinity whose orbit under $G$ corresponds to the rationals (which can be regarded as the orbit of the point at infinity under the modular group) in the classical theory of diophantine approximation.

In this paper the Hausdorff dimension of the set $W_{G,y}(\tau)$ is determined for geometrically finite groups of the first kind. Consequently, by considering the hyperboloid model of hyperbolic space, this result is shown to have a natural but non trivial interpretation in terms of quadratic forms.