Effective Bisector Estimate with Application to Apollonian Circle Packings

Abstract

Let Γ < PSL(2, C) be a geometrically finite non-elementary discrete subgroup, and let its critical exponent δ be greater than 1. We use representation theory of PSL(2, C) to prove an effective bisector counting theorem for Γ, which allows counting the number of points of Γ 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.

Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of $\operatorname {SL}_2(\mathbb {Z})$

“…If Γ < SL 2 (Z) is finitely generated with δ > 1 2 , then a version of Theorem 1.1 is known by [8] and [12] with a different interpretation of the main term (also see [22], [43], [27]). Therefore the main contribution of Theorem 1.1 lies in the groups Γ with δ ≤ 1 2 ; such groups are known to be convex cocompact.…”

From Apollonius to Zaremba: Local-global phenomena in thin orbits

“…(This asymptotic was recently refined further in Vinogradov's thesis [Vin12] and independently by Lee and Oh [LO12], giving lower order error terms.) The remainder of this subsection is devoted to explaining this spectral interpretation and highlighting some of the ideas going into the proof of (3.5).…”

On the local-global conjecture for integral Apollonian gaskets

“…The next ingredient which we require is the recent work by Vinogradov [Vin13] on effective bisector counting for such infinite volume quotients. Recall the following sub(semi)groups of G:…”

“…with s 1 , s 2 ∈ K/M , a ∈ A + and m ∈ M , corresponding to Theorem 4.14 ( [Vin13]). Let Φ, Ψ ⊂ S 2 be spherical caps and let I ⊂ R/Z be an interval.…”