Mean value results and $\Omega$-results for the hyperbolic lattice point problem in conjugacy classes

TL;DR

This paper investigates the asymptotic behavior and error estimates for counting lattice points in conjugacy classes on hyperbolic surfaces, extending classical results to a more general setting with new mean and Omega results.

Contribution

It provides new mean value and Omega results for the error term in the hyperbolic lattice point problem in conjugacy classes, including for cocompact groups and under subconvexity assumptions.

Findings

Normalized error has finite mean value in parameter t.

Integral of error over geodesic has Omega growth of X^{1/2} log log log X.

Results extend classical lattice point problem to conjugacy classes.

Abstract

For $\Gamma$ a Fuchsian group of finite covolume, we study the lattice point problem in conjugacy classes on the Riemann surface $\Gamma \backslash \mathbb{H}$. Let $\mathcal{H}$ be a hyperbolic conjugacy class in $\Gamma$ and $\ell$ the $\mathcal{H}$-invariant closed geodesic on the surface. The main asymptotic for the counting function of the orbit $\mathcal{H} \cdot z$ inside a circle of radius $t$ centered at $z$ grows like $c_{\mathcal{H}} \cdot e^{t/2}$. This problem is also related with counting distances of the orbit of $z$ from the geodesic $\ell$. For $X \sim e^{t/2}$ we study mean value and $\Omega$-results for the error term $e(\mathcal{H}, X ;z)$ of the counting function. We prove that a normalized version of the error $e(\mathcal{H}, X ;z)$ has finite mean value in the parameter $t$. Further, we prove that if $\Gamma$ is cocompact then \begin{eqnarray*} \int_{\ell} e(\mathcal{H}, X;z) d s(z) = \Omega \left( X^{1/2} \log \log \log X \right). \end{eqnarray*} We prove that the same $\Omega$-result holds for $\Gamma = {\hbox{PSL}_2( {\mathbb Z})}$ if we assume a subconvexity bound for the Epstein zeta function associated to an indefinite quadratic form in four variables. We also study pointwise $\Omega_{\pm}$-results for the error term. Our results extend the work of Phillips and Rudnick for the classical lattice problem to the conjugacy class problem.