The hyperbolic lattice counting problem in large dimensions

TL;DR

This paper studies the average behavior of the error term in the hyperbolic lattice counting problem for high-dimensional hyperbolic spaces, connecting it to quantum variance and spectral sums, and establishing lower bounds.

Contribution

It introduces new bounds and $ ext{Ω}$-results for the error term in the hyperbolic lattice counting problem in large dimensions, linking it to spectral and quantum variance concepts.

Findings

Derived upper bounds depending on quantum variance and spectral sums.

Established $ ext{Ω}$-results for the mean and second moment of the error term.

Analyzed local averages over compact sets in hyperbolic space.

Abstract

For $n\geq 3$ and $\Gamma$ a cocompact lattice acting on the hyperbolic space $\mathbb{H}^n$, we investigate the average behaviour of the error term in the circle problem. First, we explore the local average of the error term over compact sets of $\Gamma\backslash\mathbb{H}^n$. Our upper bound depends on the quantum variance and the spectral exponential sums appearing in the study of the Prime geodesic theorem. We also prove $\Omega$-results for the mean value and the second moment of the error term.