Abstract
Consider a general circle packing $\mathscr{P}$ in the complex plane $\mathbb{C}$ invariant under a Kleinian group $\Gamma$. When $\Gamma$ is convex cocompact or its critical exponent is greater than 1, we obtain an effective equidistribution for small circles in $\mathscr{P}$ intersecting any bounded connected regular set in $\mathbb{C}$; this provides an effective version of an earlier work of Oh-Shah [12]. In view of the recent result of McMullen-Mohammadi-Oh [6], our effective circle counting theorem applies to the circles contained in the limit set of a convex cocompact but non-cocompact Kleinian group whose limit set contains at least one circle. Moreover, consider the circle packing $\mathscr{P}(\mathscr{T})$ of the ideal triangle attained by filling in largest inner circles. We give an effective estimate to the number of disks whose hyperbolic areas are greater than $t$, as $t\to0$, effectivizing the work of Oh [10].
Highlights
A circle packing in the complex plane C is a countable union of circles
We introduce a measure associated with a circle packing P: Definition 2.2 (The Γ-skinning size of P)
We keep the notations from Section 2
Summary
A circle packing in the complex plane C is a countable union of circles (here a line is regarded as a circle of infinite radius). In [12], Oh and Shah considered a very general locally finite circle packing P: suppose P is invariant under a torsion-free non-elementary geometrically finite Kleinian group Γ < PSL2(C) 1. Assume P is a locally finite circle packing invariant under a geometrically finite Kleinian group Γ and with finitely many Γ-orbits. In order to prove Theorem 1.3, we need to obtain an effective estimate to the αdimensional Hausdorff measure in hyperbolic metric of neighborhoods. See [13] for related counting results
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