Effective Circle Count for Apollonian packings and Closed horospheres

2012/02/06 by Min Lee, Hee Oh, Lee, Min +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT) #math.DS #math.GT #math.NT

paper · pdf · doi:10.48550/arxiv.1202.1067

43 pages, 2 figures, To appear in GAFA

arxiv created 2012/07/18 · arxiv updated 2012/07/19

Abstract

The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume, whose fundamental group has critical exponent bigger than 1. We also discuss applications to Affine sieves. Analogous results for surfaces are treated as well.

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