An Asymptotic Local-Global Principle for Integral Kleinian Sphere Packings

Abstract: We will discuss an asymptotic local-global principle for certain integral Kleinian sphere packings. Examples of Kleinian sphere packings include Apollonian circle packings and Soddy sphere packings. Sometimes each sphere in a Kleinian sphere packing has a bend (1/radius) that is an integer. When all the bends are integral, which integers appear as bends? For certain Kleinian sphere packings, we expect that every sufficiently large integer locally represented everywhere as a bend of the packing is a bend of the packing. We will discuss ongoing work towards proving this for certain Kleinian sphere packings. This work uses the circle method, quadratic forms, and spectral theory.

number theory

Audience: researchers in the topic

Cambridge Number Theory Seminar

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