UNIFORM LATTICE POINT ESTIMATES FOR CO-FINITE FUCHSIAN GROUPS

Abstract

In this paper we obtain uniform estimates for the lattice pointproblem in the hyperbolic plane $\Bbb H$ under theassumption that the action is by a Fuchsian group $\Gamma$ which is co-finite. We fix a point $w$ from $\Bbb H$ and set $N_t(z,w)$ equal to the number of translates of $w$ by the group$\Gamma$ which lie in a geodesic ball of radius$t$ centred at a point $z$ of $\Bbb H$. The behaviour of$N_t(z,w)$ is then examined when $t$ is large and $z$ is allowed to vary over $\Bbb H$. We show that the finite quantity

$\limsup_{t\rightarrow\infty}\sup_{z\in\Bbb H} N_t (z,w) / \cosh ^2(t/2)$

depends crucially on the point $w$, and indeed can become arbitrarily large with $w$. On the other hand, for the average of this quotient we derive the estimate

\[\sup_{z\in\Bbb H} \ \frac{1}{t}\int_0^t \frac{N_{\tau}(z,w)}{\cosh ^2(\tau /2)} \, d\tau\ = \ \frac{4\pi}{| \Gamma _w | \mbox{vol}(F)} \,+ \, O\!\left(\frac{\log t}{t}\right)\]

as $t\rightarrow \infty$, where the implied constant is an explicit function of $w$. In this formula, $\mbox{vol}(F)$ is the hyperbolic volume of a Dirichletfundamental domain $F$ for $\Gamma$, and $|\Gamma _w |$ denotes the number of elements from $\Gamma$ fixing $w$. This estimate is then combined with a recent sampling theorem of K. Seip to obtain an inequality which decides whether or not the orbit $\Gamma . w$ forms a set of interpolation for a given weighted Bergman space in $\Bbb H$.