Effective bisector estimate with application to Apollonian circle packings
Abstract
Let <(2,) be a geometrically finite non-elementary discrete subgroup, and let its critical exponent δ\ be greater than 1. We use representation theory of (2,) to prove an effective bisector counting theorem for , which allows counting the number of points of \ in general expanding regions in (2,) and provides an explicit error term. We apply this theorem to give power savings in the Apollonian circle packing problem and related counting problems.
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