Effective equidistribution of circles in the limit sets of Kleinian groups

Consider a general circle packing $\mathcal{P}$ 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 $\mathcal{P}$ intersecting any bounded connected regular set in $\mathbb{C}$; this provides an effective version of an earlier work of Oh-Shah. In view of the recent result of McMullen-Mohammadi-Oh, 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 $\mathcal{P}(\mathcal{T})$ 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 $t\to 0$, effectivising the work of Oh.


Introduction
A circle packing in the complex plane C is simply a countable union of circles (here a line is regarded as a circle of infinite radius). Compared to the conventional definition of a circle packing, our definition is more general as circles are allowed to intersect each other. Given a circle packing P, we seek to estimate the number of small circles intersecting a bounded subset in C (see Figure 1 for examples).
Assume P is locally finite, i.e., for any T > 1, there are only finitely many circles in P of Euclidean curvature at most T intersecting any fixed bounded subset in C. For a bounded subset E in C and T > 1, we set N T (P, E) := #{C ∈ P : C ∩ E = ∅, Curv(C) < T }, where Curv(C) denotes the Euclidean curvature of C. As P is locally finite, N T (P, E) < ∞.
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 < PSL 2 (C) 1 . They obtained an asymptotic estimate to N T (P, E). In particular, they introduced a locally finite Borel measure ω Γ on C determined by Γ (Definition 3.10) such that under some further assumption on Γ, we have where E 1 and E 2 are any bounded Borel sets in C satisfying ω Γ (∂(E 1 )) = ω Γ (∂(E 2 )) = 0. In this paper, we extend Oh-Shah's result and provide an effective estimate to N T (P, E). To apply our theorem, we need to impose a more stringent condition on E: we require not only ω Γ (∂(E)) = 0 but the -neighborhood of ∂(E) is of small size. Sets satisfying such property will be called regular (Definition 3.14). Denote the critical exponent of Γ by δ Γ . We show the following. Theorem 1.1. Assume P is a locally finite circle packing invariant under a geometrically finite Kleinian group Γ and with finitely many Γ-orbits. When δ Γ ≤ 1, we assume further that Γ is convex cocompact. Then for any bounded connected regular set E ⊂ C, there exists η > 0, such that as T → ∞, where c > 0 is a constant depending only on Γ and P.
Denote by Λ(Γ) ⊂ C ∪ {∞} the limit set of Γ which is the set of accumulation points of an orbit of Γ in C ∪ {∞} under the linear fractional transformation action. When Γ is convex cocompact or it has no rank 2 cusps with δ Γ > 1, the measure ω Γ (E) in Theorem 1.1 equals the δ Γ -dimensional Hausdorff measure of E ∩ Λ(Γ) [17].
Circles in the limit set of a Kleinian group. Suppose Γ is convex cocompact. Consider the set of circles contained in Λ(Γ): McMullen, Mohammadi and Oh showed that if Λ(Γ) = C ∪ {∞}, there are only finitely many Γ-orbits of circles in I(Γ), and each such circle arises from a compact PSL 2 (R)-orbit (Corollary 11.3 and Theorem B.1 in [6]); this implies that I(Γ) is a locally finite circle packing with finitely many Γ-orbits and hence Theorem 1.1 applies to I(Γ). Corollary 1.2. Let Γ be a convex cocompact Kleinian subgroup with critical exponent δ Γ < 2. Assume that I(Γ) is non-empty. Then for any bounded connected regular set E ⊂ C , there exists η > 0, such that as T → ∞, where c > 0 is a constant depending only on Γ, and H δ Γ (E ∩ Λ(Γ)) is the δ Γ -dimensional Hausdorff measure of E ∩ Λ(Γ).
Circles in an ideal triangle of H 2 . Let T be an ideal triangle in the hyperbolic plane H 2 , i.e., a triangle whose sides are hyperbolic lines connecting vertices on the geometry boundary ∂H 2 . Such an ideal triangle exists and is unique up to hyperbolic isometries. Consider the circle packing P(T ) in T attained by filling in the largest inner circles (see Figure 2). We give an effective estimate to the number of disks enclosed by circles in P(T ) whose hyperbolic areas are greater than t.
Let P(T ) be the closure of P(T ). The Hausdorff dimension of P(T ), denoted by α, equals the residual dimension of an Apollonian circle packing [5]. For C ∈ P(T ), let Area hyp (C) be the hyperbolic area of the disk enclosed by C. Theorem 1.3. There exist c > 0 and η > 0, such that as t → 0, . The asymptotic formula for this counting problem without a rate was obtained by Oh in [10].
On the proof of Theorems 1.1 and 1.3. Our proof of Theorem 1.1 is built on the approach employed in [12], while providing an effective statement for each step of their arguments and combining it with the effective equidistribution results in [7]. Theorem 1.3 does not immediately follow from Theorem 1.1 since the ideal triangle is not bounded in the hyperbolic space. In order to prove Theorem 1.3, we need to obtain an effective estimate to the αdimensional Hausdorff measure in hyperbolic metric of neighborhoods We refer readers to [4] and [19] for effective counting results when P is an Apollonian circle packing. See also [13] for related counting results.
Acknowledgments. I would like to thank my advisor, Hee Oh, for suggesting the problem as a part of my thesis and for continued guidance and support. I would also like to thank Dale Winter for useful discussion, as well as comments on an earlier draft of the paper.
Let H 3 be the hyperbolic 3-space. We use the following coordinates for the upper half space model of H 3 : where j = (0, 1).
The geometric boundary ∂H 3 is the extended complex planeĈ. The group of orientation preserving isometries of H 3 is given by G. Noting that G acts transitively on H 3 and K = Stab G (j), we identify H 3 with G/K via the map [g] → gj.
Let T 1 (H 3 ) be the unit tangent bundle of H 3 , and X 0 ∈ T 1 (H 3 ) the upward unit normal vector based at j. Then T 1 (H 3 ) can be identified with G/M via the map gX 0 → [g] as M = Stab G (X 0 ).
Any circle C inĈ determines a unique totally geodesic plane in H 3 , denoted byĈ. Let C † be the unit normal bundle ofĈ. The group Stab G (Ĉ) acts transitively on bothĈ and C † . In particular, if we denote by C 0 the unit circle centered at the origin, thenĈ 0 and C † 0 can be identified with H/H ∩ K and H/M respectively as H = Stab G (Ĉ 0 ).

2.2.
Measures on Γ\G/M . Let Γ < G be a geometrically finite Kleinian group. A family of finite measures {µ x : x ∈ H 3 } is called a Γ-invariant conformal density of dimension δ µ > 0, if each µ x is a non-zero finite Borel measure on ∂H 3 satisfying for any x, y ∈ H 3 , ξ ∈ ∂H 3 and γ ∈ Γ, where ξ t is any geodesic ray tending to ξ. We denote by {ν Γ,x : x ∈ H 3 } (or simply {ν x : x ∈ H 3 }) the Patterson-Sullivan density (or PS-density), which is a Γ-invariant conformal density of dimension δ Γ with δ Γ the critical exponent of Γ. We will denote the critical exponent of Γ simply by δ when there is no room for confusion. Denote by {m x : x ∈ H 3 } the Lebesgue density, which is a G-invariant conformal density on the boundary ∂H 3 of dimension 2.
Let π : T 1 (H 3 ) → H 3 be the canonical projection map. For u ∈ T 1 (H 3 ), denote by u ± ∈ ∂H 3 the forward and the backward endpoints of the geodesic determined by u. The Hopf parametrization u → (u + , u − , s := β u − (j, π(u))) gives a homeomorphism between T 1 (H 3 ) and (∂H 3 × ∂H 3 )\{(ξ, ξ) : ξ ∈ ∂H 3 } × R. Using the identification of G/M with T 1 (H 3 ), we define the Bowen-Margulis-Sullivan measurẽ m BMS Γ and Burger-Roblin measurem BR Γ on G/M as follows: Definition 2.2 (The Γ-skinning size of P). For a circle packing P inĈ consisting of finitely many Γ-orbits, define 0 ≤ sk Γ (P) ≤ ∞ as follows: where {C i : i ∈ I} is a set of representatives of Γ-orbits in P and 3. It is shown in [17] that {ν x : x ∈ H 3 } is unique up to scalars. Using this property, we can verify that Definition 2.2 does not depend on the choice of g C i .
Theorem 2.4 (Theorem 2.4 and Lemma 3.2 in [12]). Assume that Γ is either convex cocompact or its critical exponent δ is greater than 1. Let P be a locally finite circle packing inĈ invariant under Γ with finitely many Γ-orbits. Then sk Γ (P) < ∞. Proposition 2.5. For any circle C, if Γ(C) is a locally finite circle packing consisting of infinitely many circles, then sk Γ (Γ(C)) > 0.

Effective counting for general circle packings
Throughout this section, we set Γ < PSL 2 (C) to be a torsion-free non-elementary geometrically finite Kleinian group. Let P be a locally finite circle packing consisting of finitely many Γ-orbits. Let For a subset E ⊂ C, set For s > 0, set Let E ⊂ C be a bounded Borel set. Our goal in this section is to give an effective estimate of the following counting function: where Curv(C) is the Euclidean curvature of C.
3.1. Reformulation into orbit counting problem. We built a relation between N T (P, E ) and a counting function of a Γ-orbit on H\G and prove Proposition 3.5, which is a more precise version of Proposition 3.7 in [12]. To obtain an effective estimate of N T (P, E ), it is important to understand the independence of m 0 from and the size of T in terms of , where m 0 , T and are described as Proposition 3.5.
Fix a left invariant Riemannian metric on G, which induces the hyperbolic metric on G/K = H 3 . For any > 0, set U to be the symmetric -neighborhood of e in G. For any subset W ⊂ G, denote W = U ∩ W . Define Note that there exists a constant c 0 > 0, independent of E, such that for any small > 0, E + contains the c 0 -neighborhood of E in the Euclidean metric. We fix this constant c 0 in the following.
Observe that Let C ∈ P be such that C ∩ E = ∅ and Curv(C) < T for some T > 1 c 0 . Set z 0 and r C to be the center and the radius of C respectively.
As a result, any point lying on the geodesic between w + rj and z and of Euclidean height greater than T −1 and less than 1 lies in N E + A − log T j. This proves the claim. Lemma 3.3. If E is connected, then there exists a positive integer m 0 , depending on E, such that for any T > 0, The local finiteness condition on P implies that there are only finitely many lines in P intersecting E. Denote by W T the set of all circles C ∈ P of positive curvature such thatĈ ∩ N E A − log T j = ∅ and C ∩ E = ∅. It suffices to prove that ∪ T W T consists of finitely many circles.
We claim that any circle in ∪ T W T must have radius bigger than If C ∈ ∪ T W T , it follows from the connectedness of E that E is contained in the open disc enclosed by C, and hence the radius of C is at least d E /2. Picking an arbitrary point z + sj ∈Ĉ ∩ N E A − log T j, we let w ∈ C be such that d(w, z) = d(C, z). Then d(w, z) = r C − r 2 C − s 2 ≤ s 2 /r C . As mentioned above, r C > d E /2 and hence d(w, z) ≤ 2/d E . Therefore C intersects the 2/d E -neighborhood of E non trivially, proving the claim. Now the proposition follows from the assumption that P is locally finite.
Lemmas 3.2 and 3.3 yield the following estimate: Proposition 3.5 (cf. Proposition 3.7 in [12]). Suppose E is connected. We have, for all small 0 < < 1 c 0 and for any T > 1 where m 0 is a positive integer only depending on E.

3.2.
Approximate B T (E) using HAN -decomposition. We study the shape of B T (E) in an effective way. Note that there exists c 1 > 0 depending on the metric of G, such that for all small > 0, the set K (0) := {k · 0 : k ∈ K } ⊂ C contains the disk of radius c 1 centered at 0. Set T = − log(c 1 ). We show the following inclusion: Proposition 3.6. There exists c > 0 (independent of E) such that for all sufficiently small > 0, we have for all sufficiently large T > 1 We first recall some results in [12]. (1) If a t ∈ HKa s K for s > 0, then |t| ≤ s. This in particular implies Ha t K(t) for any T > 1.
(2) Given any small > 0, we have {k ∈ K : a t k ∈ HKA + for some t > T } ⊂ K M.
In fact, the second statement of the lemma is a more precise version of Proposition 4.2 (2) in [12]. We add to the original proof the observation that K (0) contains the disk of radius c 1 centered at 0 for all small > 0.
Lemma 3.8. There exist c 2 > 1 and t 0 > 1, such that for any suffi- Proof. Fix k ∈ K and m θ ∈ M . The product map N − × A × M × N → G is a diffeomorphism at a neighborhood of e, in particular, bi-Lipschitz. Hence there exists l 1 > 1 such that for all small > 0, So k can be written as This inclusion and (3.9) yield Proof of Proposition 3.6. For all sufficiently large T > 1, it follows from Lemmas 3.7 and 3.8 that 3.3. On the measure ω Γ . Definition 3.10. Define a locally finite Borel measure ω Γ on C as follows: fixing x ∈ H 3 , for ψ ∈ C c (C), The definition of ω Γ is independent of the choice of x ∈ H 3 by the conformal properties of ν Γ,x . Lemma 3.11 (Lemma 5.2 in [12]). For any x = p + rj ∈ H 3 , and z ∈ C, we have This lemma provides another formula for ω Γ : for any ψ ∈ C c (C), 3.3.1. Relation between ω Γ and m BR . For a bounded Borel set E ⊂ C, let E + and E − be the sets defined as (3.1). For small > 0, let ψ be a non-negative smooth function in C(G) supported in U with integral one. Set Ψ ∈ C ∞ c (Γ\G) to be the Γ-average of ψ : Proposition 3.12 (cf. Lemma 5.7 in [12]). There exists c > 0 independent of E such that for all small > 0, Lemma 3.13 (Lemma 5.5 in [12]).
Proof of Proposition 3.12. Consider the following function on M AN − N ⊂ G: R E (ma t n − w n z ) = e −δt χ E (−z). We may regard R E as a function defined on G. It is shown in Lemma 5.7 in [12] that Write k −1 = m θ a t n − w n z and g = m θ 1 a t 1 n − w 1 n z 1 ∈ U . By Lemma 3.13, Using Proposition 5.4 in [12], we have which yields the proposition.

3.3.2.
Regularity criterion for ω Γ . Fix a bounded Borel set E ⊂ C. To apply the effective result in our paper, it is important to understand the difference between ω Γ (E ± ) and ω Γ (E).
Definition 3.14 (Regularity condition). We call a bounded Borel subset E ⊂ C regular if there exists 0 < p < 1 such that for all small > 0, where the implied constant depends only on E. Denote by Λ p (Γ) the set of parabolic limit points. For ξ ∈ Λ p (Γ), the rank of ξ is the rank of the abelian subgroup of Γ which fixes ξ. As Γ ⊂ PSL 2 (C), then rank(ξ) is either 1 or 2. We provide a regularity criterion for Γ with Λ p (Γ) = ∅. Proof. The proof is adapted from the proof of Proposition 7.10 in [7]. For ξ ∈ ∂H 3 , denote by s ξ = {ξ t : t ∈ [0, ∞)} the geodesic ray emanating from j toward ξ and let S(ξ t ) ∈ H 3 be the unique 2-dimensional geodesic plane which is orthogonal to s ξ at the point ξ t . Denote by b(ξ t ) the projection from j onto ∂H 3 of S(ξ t ), that is, It is shown in [16] and [17] that there exists a Γ-invariant collection of pairwise disjoint horoballs {H ξ : ξ ∈ Λ p (Γ)} for which the following holds: there exists a constant c > 1 such that for any ξ ∈ Λ(Γ) and for any t > 0, where k(ξ t ) is the rank of ξ if ξ t ∈ H ξ for some ξ ∈ Λ p (Γ) and δ otherwise. Using 0 ≤ d(ξ t , Γ(j)) ≤ t, we have for any ξ ∈ Λ(Γ) and t > 1, By standard computation in hyperbolic geometry, for any bounded set E ⊂ C, there exist c > 1 and 0 < r 0 < 1 such that for any t > − log r 0 and ξ ∈ E, where B(ξ, r) is the closed Euclidean ball in ∂H 3 of radius r. Since dω Γ = (|z| 2 + 1) δ Γ dν j , setting k 0 := max ξ ∈Λp(Γ) rank(ξ ), it follows from Now cover the -neighborhood of ∂E by {B(ξ n , 2 )}. Note that ν j is supported on Λ(Γ). For every B(ξ n , 2 ), if ξ n / ∈ Λ(Γ) but B(ξ n , 2 ) intersects Λ(Γ) nontrivially, replace it by a ball of radius 4 with center in Λ(Γ). We obtain the estimate where the constant depending only on E. Therefore, if δ > max{1, k 0 +1 2 }, then E is regular.

Conclusion.
We keep the notations from Section 2. One of our main theorems is the following: Theorem 3.21. Assume Γ is either convex cocompact or its critical exponent δ is greater than 1. Let P be a locally finite circle packing inĈ invariant under Γ with finitely many Γ-orbits. For any bounded connected regular set E in C, there exists η > 0, such that as T → ∞, we have The rest of the section is devoted to the proof of Theorem 3.21.
For ψ ∈ C ∞ (Γ\G) and l ∈ N, we consider the following L 2 -Sobolev norm of degree l: where the sum is taken over all monomials X in a fixed basis of the Lie algebra of G of order at most l and X(ψ) 2 is the L 2 (Γ\G)-norm of X(ψ). For ϕ ∈ C ∞ (Γ H \H), S l (ϕ) is defined similarly.
The key ingredient in the proof of Theorem 3.21 is the following effective equidistribution result: Theorem 3.22 ( [7]). Assume Γ is convex cocompact or its critical exponent δ is greater than 1. Suppose the natural projection Γ H \C † 0 → Γ\G/M is proper. Then there exist η 0 > 0 (depending on the spectral gap data for Γ) and l ∈ N such that for any compact subset Ω ⊂ Γ\G/M , any Ψ ∈ C ∞ (Ω) and any bounded φ ∈ C ∞ (Γ H \C † 0 ), as t → ∞, where the implied constant depends only on Ω.
(2) Strictly speaking, Theorem 3.22 in [7] is shown using the exponential mixing of geodesic flow on T 1 (Γ\H 3 ). Such an exponential mixing is provided in [18] and [14] for Γ convex cocompact and in [7] for Γ with critical exponent greater than 1.
Step 1: We first prove the theorem for P = Γ(C 0 ).
For > 0, let ψ be a non-negative function in C ∞ c (G) supported in U c 3 with integral one. Denote by Ψ ∈ C ∞ c (Γ\G) the Γ-average of ψ . For T > 1 c 0 , integrating (3.27) against Ψ , we obtain By abusing notation, we use dh to denote the Haar measures on H and H/M . We require that these two measures are compatible with the probability measure dm on M . The following defines a Haar measure on G: for g = ha r k ∈ HA + K, dg = 4 sinh r · cosh rdhdrdm j (k), where dm j (k) := dm j (kX + 0 ). Denote by dλ the unique G-invariant measure on H\G which is compatible with dg and dh.
For F ,+ T , Ψ , we have Consider the set W + T, defined in (3.25). We can rewrite it in the following form by Lemma 3.7 (1) where the set K(·) is defined as Proposition 3.6. Applying Proposition 3.6, we get: where T = − log(c 1 ) and ρ 1 > 0 is some constant. Notice that the measure e 2t dtdn is a right invariant measure of AN and [e]AN is an open subset in H\G. Hence dλ(a t n) (restricted to [e]AN ) and e 2t dtdn are constant multiplies of each other. It follows from the formula of dg that dλ(a t n) = e 2t dtdn. Besides, observe that the local finiteness of Γ(C 0 ) implies the map Γ H \C † 0 → Γ\G/M is proper. Using Theorem 3.22, we have where Ψ −z (g) = m∈M Ψ (gmn −z )dm and ρ 2 , ρ 3 > 0 are some constants. To obtain the last inequality above, we use the estimate that sup z∈E + ρ 1 S l (Ψ −z ) S l (Ψ ) because for any monomial X of order 1, we have X(Ψ −z )(g) = m∈M Ad nzm X(Ψ )(gmn −z )dm.
Since E is bounded and regular, Proposition 3.12 implies where ρ 4 , ρ 5 > 0 are some constants and p is the constant appearing in the definition of the regularity of E (Definition 3.14). Therefore, where we use the estimate S l (Ψ ) = O( −(3+l) ) since dim G=6. Consider V 1 in (3.29). Fix small such that > and it satisfies Proposition 3.6. We decompose V 1 into two parts using Proposition 3.6: where T 1 := − log(c 1 ) and ρ 7 > 0 is some constant. For V 4 , by the similar way as we get (3.31), we have for some constant ρ 8 > 0. Since E is bounded and is fixed, For V 3 , reversing the process of translating the circle counting into orbit counting, we have Adding (3.31), (3.32) and (3.33) together, we get for some constant ρ 9 > 0.
For F ,− T , Ψ , the definition of W − T, (3.25) implies the following inclusion: where T 2 is some large fixed number. Using similar argument as above, we have for some constant ρ 10 > 0. Therefore, there exists η > 0 so that as T → ∞, we have Step 2: We prove the theorem for a general circle packing. Let C be any circle in the circle packing P. Denote the radius of C by r and the center of C by z 0 . Set Then g C (C 0 ) = C. And Lemma 3.36 (Lemma 6.5 in [12]).
Lemma 3.37 (Lemma 6.7 in [12]). For any bounded Borel subset E ⊂ C, Using the above two lemmas, we obtain Since P consists of finitely many Γ-orbits, (3.38) implies our theorem.

Effective circle count for circle packing in ideal triangle of H 2
We use the upper half plane model for H 2 : For any circle packing P contained in H 2 and any t > 0, we define the following counting function: N t (P) := #{C ∈ P : Area hyp (C) > t}, where Area hyp (C) is the hyperbolic area of the disk enclosed by C.
Let T be the ideal triangle in H 2 whose three sides are given by the circles {x = ±1} and {z ∈ C : |z| = 1}. Let P(T ) be the circle packing attained by filling in the largest inner circles. This section is devoted to give an effective estimate of N t (P(T )).
Denote by P 0 the Apollonian circle packing generated by the circles {x = ±1}, {|z| = 1} and {|z−2i| = 1}. It is clear that P(T ) is a part of P 0 (see Figure 3). Let C 1 , C 2 , C 3 and C 4 be the circles {|z−(1+i)| = 1}, {|z − (−1 + i)| = 1}, {y = 0} and {y = 2} respectively. Set S i to be the reflection with respect to C i . Denote PSL 2 (C) ∩ S 1 , S 2 , S 3 , S 4 by A. We fix Γ a torsion-free finite index subgroup in A. The limit set Λ(Γ) of Γ is exactly the closure of P 0 (Proposition 2.9 in [3]). As a result, the critical exponent of Γ equals the Hausdorff dimension of the residual set of P 0 , which is denoted by α.
In fact, an elementary computation in hyperbolic geometry shows that However, it will be clear from the proof that T ( 1 4 , t) is not the right region to consider in order to obtain an effective estimate of N t (P(T )).

4.1.
Reformulation into orbit counting problem. Let C be a hyperbolic circle in H 2 . Note that C is also an Euclidean circle. Denote the Euclidean center and the Euclidean radius of C by e C and r C respectively. We have the following equivalent relation through a basic computation in hyperbolic geometry: where . Fix a circle packing P contained in T in the following. The counting function N t (P) can be reformulated as follows using (4.3): (4.4) N t (P) = #{C ∈ P : r C > Im(e C ) · β(t)}.
We introduce the following functions on H 2 : for every small > 0, For every subset E ⊂ T , define Lemma 4.7. For any small t > 0, the following holds where n(P, t) := #{C ∈ P : Area hyp (C) > t, e C / ∈ T (η, t)}.

4.2.
Number of disks in the cuspidal neighborhoods. We estimate the counting function n(P 0 , t) defined in Lemma 4.7: For all sufficiently small t > 0, we have Lemma 4.11. For all sufficiently small t > 0, we have Proof. Let C be any circle in P 0 such that C intersects {|z| ≥ t −η }.
Since P 0 is periodic, we choose C ⊂ P 0 ∩ {0 < Im z ≤ 2} so that C = C + 2k for some k ∈ N. Denote the Euclidean radius of C by r 0 . Then . It follows from Theorem 3.21 that For all sufficiently small t > 0, we have Proof. Let g 0 := ∈ PSL 2 (R). As g 0 preserves P 0 as well as the hyperbolic metric, we have where c > 0 is some constant. As a result, the estimate of N t (P 0 ∩{|z − 1| ≤ t η }) easily follows from Lemma 4.11. The estimate of N t (P 0 ∩{|z+ 1| ≤ t η }) can be verified similarly using g −1 0 .
In view of the inclusion  Recall that ν j is the PS-density at j.
Proposition 4.14. There exists N 1 ≥ 1 such that for all N ≥ N 1 , we have where E N = {z ∈ C : |z| ≥ N }.
Proof. Because the limit set Λ(Γ) of Γ is exactly the closure of P 0 , we have The stabilizer of ∞ in Γ is generated by γ 0 = 1 iy 0 0 1 for some y 0 > 0. Define the following relatively compact set in F: For any z ∈ Λ(Γ)\{∞}, there exists a unique k ∈ Z such that z ∈ γ k 0 F. This yields the estimate y 2 0 · (|k| + 1) 2 ≥ |z| 2 − | Re z| 2 ≥ |z| 2 − 1. Therefore, there exist c ≥ 1 and N 1 ≥ 1 such that for all N ≥ N 1 , we have We continue to prove the proposition. By the above inclusion, for any N ≥ N 1 , we have Proposition 4.15. There exists T 0 > 1 such that for any pair of real numbers a, b with T 0 < a < b < ∞ and any κ with κ > −2 and κ = −1, we have Proof. For a > 0 large enough, using Lemma 3.11, we have {z∈T :a<Im z<b} (Im z) κ+2α dν j (z).
Denote by π the projection map from points in {z ∈ T : a < Im z < b} to their y-coordinates. Let π * ν j be the pushforward of ν j . Letting E s = {z ∈ C : |z| > s} for s > 0, we have Now we estimate the size of the cuspidal neighborhoods under the measure (Im z) −α dω Γ . where α is the critical exponent of Γ and η is defined as (4.1).
Noting that dω Γ is a locally finite measure, we obtain the following corollary by setting κ = 0 in Proposition 4.15 : Let ψ be a non-negative function in C ∞ c (G) supported in U with integral one. Set Ψ ∈ C ∞ c (Γ\G) to be the Γ-average of ψ . We establish the following relation between m BR and (Im z) −α dω Γ : Since the product map M × A × N − × N → G has a diffeomorphic image, F ψ can be regarded as a function on G. This lemma follows from a verbatim repetition of the proof of Proposition 5.4 in [12]. Lemma 4.22. For small > 0 and any g ∈ U , k∈K/M F T (η,t) (k −1 g)dν j (k(0)) = (1+O( ·t −η )) z∈T (η,t) ± (Im z) −α dω Γ (z), where F T (η,t) := F χ T (η,t) with χ T (η,t) the characteristic function of T (η, t).
With Lemma 4.22 available, Proposition 4.19 follows from the same argument as the proof of Lemma 5.7 in [12].

Conclusion.
Theorem 4.23. There exists a constant ρ > 0, such that as t → 0, we have where dH α is the α-dimensional Hausdorff measure on P(T ).
This constraint guarantees that for every z ∈ T (η, t) + t 2η , − log(c 1 ξ) < h + t (z), where c 1 is the constant described at the beginning of Section 3.2. Set