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arXiv · 1511.00580

Lattice Point Counting in Sectors of Hyperbolic 3-space

Niko Laaksonen
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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 $1/6$.

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Niko Laaksonen. 2015-11-02. Lattice Point Counting in Sectors of Hyperbolic 3-space. https://doi.org/10.1093/qmath%2Fhax004

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