Lattice Point Counting in Sectors of Hyperbolic 3-space

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, $\mathrm{arccosh}X$, 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 $1/6$.

23 pages, 2 figures

References in corpus (1)

• Local average in hyperbolic lattice point counting