## Abstract

Let G be a finitely generated Fuchsian group of the first kind acting on the unit disk $\Delta $ and let S be the unit circle. If g is a Mobius transformation, represented by $\left( \matrix\format\c\kern.8em&\c \\ \alpha & \beta \\ \overline{\beta} & \overline{\alpha} \endmatrix \right)$ ($|\alpha|^{2}-|\beta|^{2}$ = 1) we write $\mu $(g) = 2($|\alpha|^{2}+|\beta|^{2}$). Let $\zeta \in $ S. Then there is a constant c > 0 so that for any $\eta \in $ S which is not a parabolic fixed point of G the inequality $|\eta $ - g($\zeta $)$|$ < c/$\mu $(g) has infinitely many solutions in G. This has been known for a long time (Hedlund's lemma). This can be significantly sharpened in the following manner. If G has parabolic elements let $\zeta $ be a fixed parabolic fixed point, otherwise let it be a fixed hyperbolic fixed point of G. Then there is c > 0 so that for any X > 2, $\eta \in $ S there is a solution g $\in $ G of $\aligned |\eta -g(\zeta)| & \leq c/\surd (X\mu (g))\quad (\zeta \ \text{parabolic)}\\ & \leq c/X\quad \quad \quad \quad \ \ ,(\zeta \ \text{hyperbolic}) \endaligned $ with $\mu $(g) $\leq $ X. From this we show that if w(x) is a decreasing function satisfying w(2x)/w(x) $\geq $ c > 0 then the set A = {$\eta \in $ S: $|\eta $-g($\zeta $)$|\leq $ w($\mu $(g))/$\mu $(g) is soluble for infinitely many g $\in $ G} has measure 0 or 2$\pi $ as $\Sigma $w(2$^{n}$) converges or diverges. Finally we show that the set B = {$\eta \in $ S: there is c > 0 so that $|\eta $-g($\zeta $)$|$ > c/$\mu $(g) for all g $\in $ G}, has Hausdorff dimension 1, although by the result above it has measure 0. These results are analogous to various theorems in the metrical theory of numbers, and they reduce to these if G is taken to be the modular group. The proofs involve a close study of the geometry of the action of G on $\Delta $.