[Paper Review] Effective bisector estimate with application to Apollonian circle packings

Ilya Vinogradov|arXiv (Cornell University)|Apr 24, 2012

Mathematical Dynamics and Fractals22 references4 citations

TL;DR

This paper establishes an effective bisector counting theorem for geometrically finite, non-elementary subgroups $Γ < \mathrm{PSL}(2,\mathbf{C})$ with critical exponent $\delta > 1$, using representation theory of $\mathrm{PSL}(2,\mathbf{C})$ to derive explicit error terms. The method enables power savings in the Apollonian circle packing problem and related counting problems by leveraging $K$-type decompositions of complementary series representations and spectral analysis of matrix coefficients.

ABSTRACT

Let Γ

Motivation & Objective

• To develop an effective counting theorem for orbits of discrete subgroups $\Gamma < \mathrm{PSL}(2,\mathbf{C})$ in expanding regions defined by bisectors.
• To address the lack of effective error terms in non-lattice, infinite covolume settings for hyperbolic counting problems.
• To apply the main result to achieve power savings in the Apollonian circle packing problem.
• To extend the scope of effective counting beyond balls and sectors to general expanding sets in $\mathrm{PSL}(2,\mathbf{C})$.
• To establish a framework for effective counting in geometrically finite, non-lattice subgroups using spectral theory and $K$-type decomposition.

Proposed method

• Utilize the representation theory of $\mathrm{PSL}(2,\mathbf{C})$, particularly complementary series representations, to analyze $L^2(\Gamma\backslash G)$.
• Perform a $K$-type decomposition of complementary series representations to control matrix coefficients and spectral projections.
• Apply spectral decomposition and conjugation invariance to reduce the counting problem to estimates on matrix coefficients.
• Use the $KA^+K$ decomposition and Lie algebra structure to parametrize group elements and control growth in $a$ and $a'$ variables.
• Implement smoothing via partitions of unity and approximate sums over $\beta$-orbits using Fourier coefficient estimates.
• Apply the main theorem in simplified form to derive asymptotic formulas with explicit error terms, balancing contributions from different error sources.

Experimental results

Research questions

• RQ1 Can an effective bisector counting theorem be established for non-lattice, geometrically finite subgroups $\Gamma < \mathrm{PSL}(2,\mathbf{C})$ with $\delta > 1$?
• RQ2 What is the optimal error term in the counting of $\Gamma$-orbits in expanding bisector regions, and can it be made effective using spectral methods?
• RQ3 How can the spectral theory of $\mathrm{PSL}(2,\mathbf{C})$ be leveraged to achieve power savings in the Apollonian circle packing problem?
• RQ4 To what extent can the method of $K$-type decomposition for complementary series representations be generalized to non-lattice settings?
• RQ5 Can the error terms from smoothing and spectral decomposition be balanced to yield a non-trivial power-saving bound in counting problems?

Key findings

• An effective bisector counting theorem is established with an explicit error term of order $O(T^{\frac{10\delta + s_1}{11} + \varepsilon})$ for $\delta > 1$, improving upon previous ineffective or asymptotic results.
• The leading term in the counting formula is proportional to $T^\delta$, consistent with the exponential growth rate dictated by the critical exponent $\delta$, and involves integrals over the boundary $\partial\mathbf{H}^3$ weighted by $\delta$-dimensional density.
• Power savings are achieved in the Apollonian circle packing problem by applying the main theorem to count circles in ideal triangles, yielding an error term smaller than any power of $T$ in the leading term.
• The method balances two error contributions: $VT^\delta$ and $T^{\frac{10\delta + s_1}{11} + \varepsilon} V^{-4 - 2(\frac{15}{11} + 2) + \varepsilon}$, and equating them at $V = T^{-(\delta - s_1)/129}$ yields optimal trade-off.
• The error from bounding the function $f$ via smoothing is shown to be negligible compared to the main error terms, justifying its neglect in the asymptotic estimate.
• The framework is extended to ideal triangle counting by modifying the characteristic function to restrict to a fundamental domain $G_4$, and the resulting formula matches the main theorem with adjusted measures.