@article{Kontorovich_Lutsko_2024, title={Effective Counting in Sphere Packings}, volume={2}, url={https://jamathr.org/index.php/jamr/article/view/Vol-2Issue-1Paper-2}, abstractNote={<div class="page" title="Page 3"> <div class="layoutArea"> <div class="column"> <p>Given a Zariski-dense, discrete group, Γ, of isometries acting on (n + 1)- dimensional hyperbolic space, we use spectral methods to obtain a sharp asymptotic formula for the growth rate of certain Γ-orbits. In particular, this allows us to obtain a best-known effective error rate for the Apollonian and (more generally) Kleinian sphere packing counting problems, that is, counting the number of spheres in such with radius bounded by a growing parameter. Our method extends the method of Kontorovich [Kon09], which was itself an extension of the orbit counting method of Lax-Phillips [LP82], in two ways. First, we remove a compactness condition on the discrete subgroups considered via a technical cut- off and smoothing operation. Second, we develop a coordinate system which naturally corresponds to the inversive geometry underlying the sphere counting problem, and give structure theorems on the arising Casimir operator and Haar measure in these coordinates.</p> </div> </div> </div>}, number={1}, journal={Journal of the Association for Mathematical Research}, author={Kontorovich, Alex and Lutsko, Christopher}, year={2024}, month={Mar.}, pages={15–52} }