Effective bisector estimate with application to Apollonian circle packings

Abstract

Let Gamma < PSL(2, C) be a geometrically finite non-elementary discrete subgroup, and let its

critical exponent delta be greater than 1. We use representation theory of PSL(2, C) to prove an

effective bisector counting theorem for Gamma, which allows counting the number of points of Gamma in

general expanding regions in PSL(2, C) and provides an explicit error term. We apply this

theorem to give power savings in the Apollonian circle packing problem and related counting

problems.