An application of metric diophantine approximation in hyperbolic space to quadratic forms
Here is the author's abstract: ``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 nontrivial interpretation in terms of quadratic forms.'' Specifically, if we let \(\delta (G)\) be the exponent of convergence of \(G\), the author has: Theorem 1. For \(\tau > 1\), \[ \dim W_{G,y} (\tau) \leq \delta (G)/ \tau. \] Theorem 2. If the group \(G\) is of the first kind, then for \(\tau > 1\) \[ \dim W_{G,y} (\tau) = k/ \tau. \] Note that for groups of the first kind \(\delta (G) = k\); so that the inequality in Theorem 1 is sharp. As the author notes: ``... the proofs of the theorems follow on using essentially the same arguments as in the Fuchsian case \((k = 1)\); and will therefore be merely sketched. ... The main purpose of this article is to give an interpretation of Theorem 2 in terms of quadratic forms.'' This ``interpretation'' is too technical to be given here. It concerns a natural, though somewhat specific quadratic form derived from the Euclidean metric on \(\mathbb{R}^n\).
scientific article; zbMATH DE number 722041