Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds

By Alex Kontorovich, H. Oh. 2008

137 citations recorded by the source corpus.

We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples. The main ingredient of our proof is the effective equidistribution of expanding horospheres on geometrically finite hyperbolic 3-manifolds under the assumption that the critical exponent of its fundamental group exceeds one.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/df6ce777fb00d8c4cb81dd245db6a3054d504aac

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits (2012, background): “t = 1:30568::: (3.4) From such considerations, Boyd [Boy82] was able to conclude that logN G (T) logT !; as T!1. To rene this crude estimate to an asymptotic formula for N G (T), the author and Oh [KO11] established a \spectral interpretation" for L G , proving: N G (T) ˘cT; (3.5) FROM APOLLONI”
Almost prime Pythagorean triples in thin orbits (2010, methodology): “This is the same as [21], Proposition 6.”
Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars (2014, unknown): “The stronger result      a ~ a c dimH ( ) ( ( )) was proved in [35].”
Notes on Thin Matrix Groups (2012, methodology): “Thanks to the referee for pointing me to the relevant applications in [Gromov and Guth 2012].”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, methodology): “7 was proved in [35], [34] and [41] by a different method.”
Generalization of Selberg's 3/16 Theorem and Affine Sieve (2009, methodology): “g Fermi coordinates. In particular, by combining the H3 analogue of Theorem 1.2 with [37] and Theorem 6.3 in [4] we obtain the following theorem which has applications to integral Apollonian packings [10, 15, 30]. 10 JEANBOURGAIN,ALEXGAMBURD,ANDPETERSARNAK Theorem2.1.Let Λ be a geometrically-ﬁnite s”