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Statistical Regularity of Apollonian Gaskets 11 STATISTICAL REGULARITY OF APOLLONIAN GASKETS XIN ZHANG Abstract. Apollonian gaskets are formed by repeatedly filling the gaps between three mutually tangent circles with further tangent circles. In this paper we give explicit formulas for the the limiting pair correlation and the limiting nearest neighbor spacing of centers of circles from a fixed Apollonian gasket. These are corollaries of the convergence of moments that we prove. The input from ergodic theory is an extension of Mohammadi-Oh's Theorem on the equidisribution of expanding horospheres in infinite volume hyperbolic spaces. 1. Introduction 1.1. Introduction to the problem and statement of results. Apollonian gaskets, named after the ancient Greek mathematician Apollonius of Perga (200 BC), are fractal sets formed by starting with three mutually tangent circles and iteratively inscribing new circles into the curvilinear triangular gaps (see Figure 1). Figure 1. Construction of an Apollonian gasket The last 15 years have overseen tremendous progress in understanding the structure of Apollonian gaskets from different viewpoints, such as number theory and geometry [16], [15], [10], [11], [20], [25]. In the geometric direction, generalizing a result of [20], Hee Oh and Nimish Shah proved the following remarkable theorem concerning the growth of circles. Place an Apollonian gasket P in the complex plane C. Let Pt be the set of circles −t from P with radius greater than e and let Ct be the set of centers from Pt. Oh-Shah proved: Theorem 1.1 (Oh-Shah, Theorem 1.6, [25]). There exists a finite Borel measure v supported on P, such that for any compact region E ⊂ C with boundary @E empty or 1 2 XIN ZHANG piecewise smooth (see Figure 2), the cardinality N(E; t) of the set Ct \ E, satisfies N(E; t) lim = v(E); t!1 eδt where δ ≈ 1:305688 [22] is the Hausdorff dimension of any Apollonian gasket. Figure 2. A region E with piecewise smooth boundary Theorem 1.1 gives a satisfactory explanation on how circles are distributed in an Apollonian gasket in large scale. In this paper we study some questions concerning the fine scale distribution of circles, for which Theorem 1.1 yields little information. For example, one such question is the following. t Question 1.2. Fix ξ > 0. How many circles in Pt are within distance ξ=e of a random circle in Pt? Here by distance of two circles we mean the Euclidean distance of their centers. Question 1.2 is closely related to the pair correlation of circles. In this article, we study the pair correlation and the nearest neighbor spacing of circles, which concern the fine structures of Apollonian gaskets. In particular, Theorem 1.3 gives an asymptotic formula for one half of the expected number of circles in Question 1.2, as t ! 1. Let E ⊂ C be an open set with @E empty or piecewise smooth as in Theorem 1.1, and with v(E) > 0 (if E \ P 6= ; then v(E) > 0). This is our standard assumption for E throughout this paper. The pair correlation function PE;t on the growing set Ct is defined as 1 X t PE;t(ξ) := 1fe jp − qj < ξg; (1.1) 2#fCt \ Eg p;q2Ct\E q6=p where ξ 2 (0; 1) and jp − qj is the Euclidean distance between p and q in C. We have a factor 1=2 in the definition (1.1) because we only want to count each pair of points once. STATISTICAL REGULARITY OF APOLLONIAN GASKETS 3 For any p 2 Ct, let dt(p) = minfjq −pj : q 2 Ct; q 6= pg. The nearest neighbor spacing function QE;t is defined as 1 X t QE;t(ξ) = 1fe dt(p) < ξg (1.2) #fCt \ Eg p2Ct\E For simplicity we abbreviate PE;t;QE;t as Pt;Qt if E = C. It is noteworthy that in both definitions (1.1) and (1.2), we normalize distance by multiplying by et. The reason can be seen in two ways. First, Theorem 1.1 implies that a random circle in Ct has radius e−t, so a random pair of nearby points (say, the centers of two tangent −t −t circles) from Ct has distance e , thus e is the right scale to measure the distance of two nearby points in Ct. The second explanation is more informal: if N points are randomly distributed in the unit interval [0; 1], then a random gap is of the scale N −1; more generally, if N points are randomly distributed in a compact manifold of dimension n, the distance between a random pair of nearby points should be of the −1=n scale N . In our situation, as t ! 1, the set Ct converges to P , where P has δt Hausdorff dimension δ ≈ 1:305688. From Theorem 1.1, we know that #Ct e , so our scaling e−t agrees with the heuristics that the distance between two random nearby δt − 1 −t points in Ct should be (e ) δ = e . Before stating our main results, we introduce terminology. It is convenient for us to work with the upper half-space model of the hyperbolic 3-space H3: 3 H = fz + rj : z = x + yi 2 C; r 2 Rg: We identify the boundary @H3 of H3 with C [ f1g. For q = x + yi + rj 2 H3, we define <(q) = x + yi and =(q) = r. Let G = P SL(2; C) be the group of orientation-preserving isometries of H3. We choose a discrete subgroup Γ < P SL(2; C) whose limit set Λ(Γ) = P such that Γ acts transitively on circles from P. It follows from Corollary 1.3, [6] that Γ is geometrically finite. Without loss of generality, we can assume that the bounding circle of P is C(0; 1), where C(z; r) ⊂ C is the circle centered at z with radius r. Let S ⊂ H3 be the hyperbolic geodesic plane with @S = C(0; 1), and H < P SL(2; C) be the stabilizer of S. As an isometry on H3, each g 2 P SL(2; C) sends S to a geodesic plane, which is either a vertical plane or a hemisphere in the upper half-space model of H3. We define 3 continuous maps q : G ! H , q< : G ! Cb as follows: ( the apex of g(S); if 1 62 g(@S); q(g) := (1.3) 1; if 1 2 g(@S) ( <(q(g)); if 1 62 g(@S); q (g) := (1.4) < 1; if 1 2 g(@S): 4 XIN ZHANG 3 ∗ 3 We further define a few subsets of H . Let Bξ := fz 2 C : jzj < ξg and let Bξ ⊂ H be the “infinite chimney" with base Bξ, where for any Ω ⊂ C, Ω∗ := fz + rj : z 2 Ω; r 2 (1; 1)g: (1.5) 3 Let Cξ be the cone in H : r 1 C := z + rj 2 3 : > ; 0 < r ≤ 1 : (1.6) ξ H jzj ξ Now we can state our main theorems. Theorem 1.3 (limiting pair correlation). For any open set E ⊂ C with @E empty or piecewise smooth, there exists a continuously differentiable function P independent of E, supported on [c; 1) for some c > 0, such that lim PE;t(ξ) = P (ξ): t!1 The derivative P 0 of P is explicitly given by Z −1 −1 δ 0 δ X jq<(h γ )j PS P (ξ) = PS δ+1 dµH (h): 2µH (ΓH nH) h2ΓH nH ξ γ2γH n(Γ−ΓH ) −1 −1 ∗ q(h γ )2Bξ [Cξ PS PS Here ΓH := Γ \ H, and µH is a Patterson-Sullivan type measure on H. Besides µH , PS BR BMS we will also encounter other conformal measures µN ; w; m ; m , which are built PS on the Patterson-Sullivan densities. The measure µN is a Patterson-Sullivan type 1 z measure on the horospherical group N := n = : z 2 , w is the pullback z 0 1 C PS BR BMS measure of µN on C under the identification z ! nz, and m ; m are the Burger- Roblin, Bowen-Margulis-Sullivan measures. We will have a detailed discussion of these measures in Section 4. See Figure 3 and Figure 5 for some numerical evidence for Theorem 1.3. Let P(θ1; θ2) be the unique Apollonian gasket determined by the four mutually tangent circles C0;C1;C2;C3, where C0 = C(0; 1) is the bounding circle, and C1;C2;C3 are θ1i θ2i tangent to C0 at 1; e , e . Figure 3, Figure 4 and Figure 5 are based on the gasket 1:8π 3:7π P( 3 ; 3 ). Figure 6 suggests that the limiting pair correlations for different Apollo- nian gaskets are the same. The reason is twofold. First, for a fixed gasket, the limiting pair correlation locally looks the same everywhere. Second, one can take any Apollo- nian gasket to any other one by a M¨obiustransformation, which locally looks like a dilation combined with a rotation, and it is an elementary exercise to check that the limiting pair correlation is invariant under these motions. STATISTICAL REGULARITY OF APOLLONIAN GASKETS 5 Figure 3. The plot for Pt with various t's Figure 4. Pair correlation for the whole plane, half plane and the first quadrant 6 XIN ZHANG 0 Figure 5. The empirical derivative Pt (ξ) for different t, with step=0.1 Figure 6. Pair correlation functions for different Apollonian gaskets STATISTICAL REGULARITY OF APOLLONIAN GASKETS 7 Theorem 1.4 (limiting nearest neighbor spacing). There exists a continuous function Q independent of E, supported on [c; 1) for some c > 0, such that lim QE;t(ξ) = Q(ξ): (1.7) t!1 The formula for Q is explicitly given by Z Z 1 δ −δt −1 ∗ PS Q(ξ) = 1 − PS e 1f#q(a−th (Γ − ΓH )) \ Bξ = 0gdtdµH (h): µH (ΓH nH) ΓH nH 0 (1.8) See Figure 7 for numerical evidence.
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