Effective Counting in Sphere Packings



Orbital counting, Spectral theory, Automorphic representations


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.




How to Cite

Kontorovich, A., & Lutsko, C. (2024). Effective Counting in Sphere Packings. Journal of the Association for Mathematical Research, 2(1), 15–52. Retrieved from https://jamathr.org/index.php/jamr/article/view/Vol-2Issue-1Paper-2