2015/11/02 by Niko Laaksonen
Mathematics · #Combinatorics #Geodesic #Geometric and Algebraic Topology #Geometry and complex manifolds #Hyperbolic space #Hyperplane #Lattice (music) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pointwise #Pure mathematics #Submanifold #math.NT #msc:11F72 #msc:11N36
paper · pdf · doi:10.1093/qmath/hax004
published as Q. J. Math, 68, Issue 3, 2017, pp. 891-922 · 23 pages, 2 figures
arxiv created 2015/11/02 · openalex publication_date 2017/01/30 · arxiv updated 2017/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let Γ be a cocompact discrete subgroup of PSL2(C) and denote by H the three-dimensional upper half-space. For a p∈H, we count the number of points in the orbit Γp, according to their distance, arccoshX, 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(X3/2) 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(X1+ε) on an 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.