Lattice Point Counting in Sectors of Hyperbolic 3-space

TL;DR

This paper studies counting lattice points in hyperbolic 3-space sectors, providing new error bounds and applying large sieve techniques to improve average-case estimates.

Contribution

It extends pointwise error analysis to three dimensions and introduces large sieve inequalities for improved average error bounds.

Findings

Established a pointwise error term of O(X^{3/2})

Derived an average error bound of O(X^{1+ε}) using large sieve inequalities

Explained the limitations of radial average improvements on error bounds

Abstract

Let $\Gamma$ be a cocompact discrete subgroup of $\mathrm{PSL}_{2}(\mathbb{C})$ and denote by $\mathcal{H}$ the three dimensional upper half-space. For a $p\in\mathcal{H}$, we count the number of points in the orbit $\Gamma p$, according to their distance, $\operatorname{arccosh} X$, from a totally geodesic hyperplane. The main term in $n$ dimensions was obtained by Herrmann for any subset of a totally geodesic submanifold. We prove a pointwise error term of $O(X^{3/2})$ by extending the method of Huber and Chatzakos-Petridis to three dimensions. By applying Chamizo's large sieve inequalities we obtain the conjectured error term $O(X^{1+\epsilon})$ on average in the spatial aspect. We prove a corresponding large sieve inequality for the radial average and explain why it only improves on the pointwise bound by $1/6$.