apollonian circle packings: dynamics and number theory

Japan. J. Math. 9, 69–97 (2014)DOI: 10.1007/s11537-014-1384-6

Apollonian circle packings: dynamics andnumber theory?

Hee Oh

Received: 5 December 2013 / Accepted: 21 January 2014Published online: 20 February 2014© The Mathematical Society of Japan and Springer Japan 2014

Communicated by: Toshiyuki Kobayashi

Abstract. We give an overview of various counting problems for Apollonian circle packings,which turn out to be related to problems in dynamics and number theory for thin groups. Thissurvey article is an expanded version of my lecture notes prepared for the 13th Takagi Lecturesgiven at RIMS, Kyoto in the fall of 2013.

Keywords and phrases: Apollonian circle packing, expander, thin group, geometrically finitegroup, equidistribution

Mathematics Subject Classification (2010): 11N45, 37F35, 22E40

Contents

1. Counting problems for Apollonian circle packings ..................................................... 702. Hidden symmetries and orbital counting problem....................................................... 753. Counting, mixing, and the Bowen–Margulis–Sullivan measure .................................... 784. Integral Apollonian circle packings .......................................................................... 845. Expanders and sieve............................................................................................... 88

? This article is based on the 13th Takagi Lectures that the author delivered at Research Institutefor Mathematical Sciences, Kyoto University on November 16 and 17, 2013.

H. OH

Mathematics Department, Yale University, New Haven, CT 06520, USAand Korea Institute for Advanced Study, Seoul, Republic of Korea(e-mail: [email protected])

70 H. Oh

1. Counting problems for Apollonian circle packings

An Apollonian circle packing is one of the most of beautiful circle packingswhose construction can be described in a very simple manner based on an oldtheorem of Apollonius of Perga:

Theorem 1.1 (Apollonius of Perga, 262–190 BC). Given 3 mutually tangentcircles in the plane, there exist exactly two circles tangent to all three.

Proof. We give a modern proof, using the linear fractional transformations ofPSL2.C/ on the extended complex plane OC D C [ f1g, known as Möbiustransformations: �

a b

c d

�.z/ D az C b

cz C d;

where a; b; c; d 2 C with ad � bc D 1 and z 2 C [ f1g. As is well known,a Möbius transformation maps circles in OC to circles in OC, preserving anglesbetween them (In the whole article, a line in C is treated as a circle in OC). Inparticular, it maps tangent circles to tangent circles.

For given three mutually tangent circles C1; C2; C3 in the plane, denote byp the tangent point between C1 and C2, and let g 2 PSL2.C/ be an elementwhich maps p to 1. Then g maps C1 and C2 to two circles tangent at 1, thatis, two parallel lines, and g.C3/ is a circle tangent to these parallel lines. Inthe configuration of g.C1/, g.C2/, g.C3/ (see Fig. 1), it is clear that there areprecisely two circles, say, D and D0 tangent to all three g.Ci /, 1 � i � 3.Using g�1, which is again a Möbius transformation, it follows that g�1.D/ andg�1.D0/ are precisely those two circles tangent to C1, C2, C3. �

Fig. 1. Pictorial proof of the Apollonius theorem

In order to construct an Apollonian circle packing, we begin with four mu-tually tangent circles in the plane (see Fig. 2 for possible configurations) andkeep adding newer circles tangent to three of the previous circles provided byTheorem 1.1. Continuing this process indefinitely, we arrive at an infinite circlepacking, called an Apollonian circle packing.

Apollonian circle packings: dynamics and number theory 71

Fig. 2. Possible configurations of four mutually tangent circles

Figure 3 shows the first few generations of this process, where each circle islabeled with its curvature (D the reciprocal of its radius) with the normalizationthat the greatest circle has radius one.

3

Fig. 3. First few generations

If we had started with a configuration containing two parallel lines, we wouldhave arrived at an unbounded Apollonian circle packing as in Fig. 4. There arealso other unbounded Apollonian packings containing either only one line or noline at all; but it will be hard to draw them in a paper with finite size, as circleswill get enormously large only after a few generations.

Fig. 4. Unbounded Apollonian circle packing

For a bounded Apollonian packing P , there are only finitely many circlesof radius bigger than a given number. Hence the following counting function iswell-defined for any T > 0:

NP.T / WD #fC 2 P W curv.C / � T g:

72 H. Oh

Question 1.2. � Is there an asymptotic formula of NP.T / as T ! 1?� If so, can we compute?

The study of this question involves notions related to metric properties of theunderlying fractal set called a residual set:

Res.P/ WD[

C 2P

C ;

i.e., the residual set of P is the closure in C of the union of all circles in P .The Hausdorff dimension of the residual set of P is called the residual

dimension of P , which we denote by ˛. The notion of the Hausdorff dimensionwas first given by Hausdorff in 1918. To explain its definition, we first recall thenotion of the Hausdorff measure (cf. [36]):

Definition 1.3. Let s � 0 and F be any subset of Rn. The s-dimensional Haus-dorff measure of F is defined by

H s.F / WD lim�!0

�inf

nXd.Bi /

s W F �[

i

Bi ; d.Bi / < �o�;

where d.Bi / is the diameter of Bi .

For s D n, it is the usual Lebesgue measure of Rn, up to a constant multiple.It can be shown that as s increases, the s-dimensional Hausdorff measure ofF will be 1 up to a certain value and then jumps down to 0. The Hausdorffdimension of F is this critical value of s:

dimH .F / D supfs W H s.F / D 1g D inffs W H s.F / D 0g:In fractal geometry, there are other notions of dimensions which often have

different values. But for the residual set of an Apollonian circle packing, theHausdorff dimension, the packing dimension and the box dimension are allequal to each other [55].

We observe

� 1 � ˛ � 2;� ˛ is independent of P: any two Apollonian packings are equivalent to each

other by a Möbius transformation which maps three tangent points of onepacking to three tangent points of the other packing.

� The precise value of ˛ is unknown, but approximately, ˛ D 1:30568.8/ dueto McMullen [38].

In particular, Res.P/ is much bigger than a countable union of circles (as˛ > 1), but not too big in the sense that its Lebesgue area is zero (as ˛ < 2).

The first counting result for Apollonian packings is due to Boyd in 1982 [8]:


Theorem 1.4 (Boyd).

limT !1

logNP.T /

logTD ˛:

Boyd asked in [8] whether NP.T / � c � T ˛ as T ! 1, and wrote that hisnumerical experiments suggest this may be false and perhaps

NP.T / � c � T ˛.logT /ˇ

might be more appropriate.However it turns out that there is no extra logarithmic term:

Theorem 1.5 (Kontorovich–O. [30]). For a bounded Apollonian packing P ,there exists a constant cP > 0 such that

NP.T / � cP � T ˛ as T �! 1:

Theorem 1.6 (Lee–O. [32]). There exists � > 0 such that for any boundedApollonian packing P ,

NP.T / D cP � T ˛ CO.T ˛��/:

Vinogradov [58] has also independently obtained Theorem 1.6 with a weakererror term.

For an unbounded Apollonian packing P , we have NP.T / D 1 in gen-eral; however we can modify our counting question so that we count only thosecircles contained in a fixed curvilinear triangle R whose sides are given by threemutually tangent circles.

SettingNR.T / WD #fC 2 R W curv.C / � T g < 1;

we have shown:

74 H. Oh

Theorem 1.7 (O.–Shah [44]). For a curvilinear triangle R of any Apollonianpacking P , there exists a constant cR > 0 such that

NR.T / � cR � T ˛ as T �! 1:

Going even further, we may ask if we can describe the asymptotic distri-bution of circles in P of curvature at most T as T ! 1. To formulate thisquestion precisely, for any bounded region E � C, we set

NP.T;E/ WD #fC 2 P W C \E ¤ ;; curv.C / � T g:Then the question on the asymptotic distribution of circles in P amounts to

searching for a locally finite Borel measure !P on the plane C satisfying that

limT !1

NT .P; E/

T ˛D !P.E/

for any bounded Borel subset E � C with negligible boundary.Noting that all the circles in P lie on the residual set of P , any Borel mea-

sure describing the asymptotic distribution of circles of P must be supportedon Res.P/.

Theorem 1.8 (O.–Shah [44]). For any bounded Borel E � C with smoothboundary,

NP.T;E/ � cA � H ˛P.E/ � T ˛ as T �! 1;

where H ˛P denotes the ˛-dimensional Hausdorff measure of the set Res.P/

and 0 < cA < 1 is a constant independent of P .

In general, dimH .F / D s does not mean that the s-dimensional Hausdorffmeasure H s.F / is non-trivial (it could be 0 or 1). But on the residual setRes.P/ of an Apollonian packing, H ˛

P is known to be locally finite and its sup-port is precisely Res.P/ by Sullivan [57]; hence H ˛

P.E/ < 1 for E boundedand 0 < H ˛

P.E/ if Eı \ Res.P/ ¤ ;.Though the Hausdorff dimension and the packing dimension are equal to

each other for Res.P/, the packing measure is locally infinite ([57], [37]) whichindicates that the metric properties of Res.P/ are subtle.

Theorem 1.8 says that circles in an Apollonian packing P are uniformlydistributed with respect to the ˛-dimensional Hausdorff measure on Res.P/:for any bounded Borel subsets E1; E2 � C with smooth boundaries and withEı

2 \ Res.P/ ¤ ;,

limT !1

NP.T;E1/

NP.T;E2/D H ˛

P.E1/

H ˛P.E2/

:


Apollonian constant

Observe that the constant cA in Theorem 1.8 is given by

cA D limT !1

NP.T;E/

T ˛ � H ˛P.E/

for any Apollonian circle packing P and any E with Eı \ Res.P/ ¤ ;. Inparticular,

cA D limT !1

NP.T /

T ˛ � H ˛P.Res.P//

for any bounded Apollonian circle packing P .

Definition 1.9. We propose to call 0 < cA < 1 the Apollonian constant.

Problem 1.10. What is cA?

Whereas all other terms in the asymptotic formula of Theorem 1.7 can bedescribed using the metric notions of Euclidean plane, our exact formula of cAinvolves certain singular measures of an infinite-volume hyperbolic 3 manifold,indicating the intricacy of the precise counting problem.

2. Hidden symmetries and orbital counting problem

Hidden symmetries

The key to our approach of counting circles in an Apollonian packing lies in thefact that

An Apollonian circle packing has lots of hidden symmetries.

Explaining these hidden symmetries will lead us to explain the relevance of thepacking with a Kleinian group, called the (geometric) Apollonian group.

Fix 4 mutually tangent circles C1; C2; C3; C4 in P and consider their dualcircles OC1; : : : ; OC4, that is, OCi is the unique circle passing through the threetangent points among Cj ’s for j ¤ i . In Fig. 5, the solid circles representCi ’s and the dotted circles are their dual circles. Observe that inverting withrespect to a dual circle preserves the three circles that it meets perpendicularlyand interchanges the two circles which are tangent to those three circles.

Definition 2.1. The inversion with respect to a circle of radius r centered at amaps x to aC r2

jx�aj2 .x � a/. The group Möb. OC/ of Möbius transformations inOC is generated by inversions with respect to all circles in OC.

76 H. Oh

Fig. 5. Dual circles

The geometric Apollonian group A WD AP associated to P is generatedby the four inversions with respect to the dual circles:

A D h�1; �2; �3; �4i < Möb. OC/;where �i denotes the inversion with respect to OCi . Note that PSL2.C/ is a sub-group of Möb. OC/ of index two; we will write Möb. OC/ D PSL2.C/

˙. The Apol-lonian group A is a Kleinian group (D a discrete subgroup of PSL2.C/

˙) andsatisfies

� P D S4iD1 A .Ci /, that is, inverting the initial four circles in P with re-

spect to their dual circles generates the whole packing P;� Res.P/ D ƒ.A / where ƒ.A / denotes the limit set of A , which is the set

of all accumulation points of an orbit A .z/ for z 2 OC.

In order to explain how the hyperbolic geometry comes into the picture, itis most convenient to use the upper-half space model for hyperbolic 3 spaceH3: H3 D f.x1; x2; y/ W y > 0g. The hyperbolic metric is given by ds Dq

dx21Cdx2

2Cdy2

yand the geometric boundary @1.H3/ is naturally identified with

OC. Totally geodesic subspaces in H3 are vertical lines, vertical circles, verticalplanes, and vertical hemispheres.


The Poincaré extension theorem gives an identification Möb. OC/ with theisometry group Isom.H3/. Since Möb. OC/ is generated by inversions with re-spect to circles in OC, the Poincaré extension theorem is determined by the cor-respondence which assigns to an inversion with respect to a circle C in OC theinversion with respect to the vertical hemisphere in H3 above C . An inversionwith respect to a vertical hemisphere preserves the upper half space, as well asthe hyperbolic metric, and hence gives rise to an isometry of H3.

The Apollonian group A D AP , now considered as a discrete subgroupof Isom.H3/, has a fundamental domain in H3, given by the exterior of thehemispheres above the dual circles to P . In particular, A nH3 is an infinitevolume hyperbolic 3-manifold and has a fundamental domain with finitely manysides; such a manifold is called a geometrically finite manifold.

Connection with the Whitehead link

The Apollonian manifold A nH3 can also be constructed from the Whiteheadlink complement. To explain the connection, consider the group, say, A � gen-erated by 8 inversions with respect to four mutually tangent circles as well astheir four dual circles. Then the group A � has a regular ideal hyperbolic octahe-dron as a fundamental domain in H3, and is commensurable to the Picard groupPSL2.ZŒi �/, up to a conjugation, which is a lattice in PSL2.C/. The quotientorbifold A �nH3 is commensurable to the Whitehead link complement S3 �W(see Fig. 6). In this finite volume 3-manifold S3 � W , we have a triply punc-tured sphere (corresponding a disk in S3 spanning one component of W andpierced twice by the other component), which is totally geodesic and whosefundamental group is conjugate to the congruence subgroup �.2/ of PSL2.Z/

of level 2. If we cut the manifold S3 �W open along this totally geodesic sur-face �.2/nH2, we get a finite volume hyperbolic manifold with totally geodesicboundary, whose fundamental group is the Apollonian group A . We thank Curt

Fig. 6. Whitehead link

78 H. Oh

McMullen for bringing this beautiful relation with the Whitehead link to ourattention.

Orbital counting problem in PSL2.R/n PSL2.C/

Observe that the number of circles in an Apollonian packing P of curvature atmost T is same as the number of the vertical hemispheres above circles in Pof Euclidean height at least T �1. Moreover for a fixed bounded region E in C,NP.T;E/ is same as the number of the vertical hemispheres above circles inP which intersects the cylindrical region

ET WD f.x1; x2; y/ 2 H3 W x1 C ix2 2 E; T �1 � y � r0g; (2.2)

where r0 > 0 is the radius of the largest circle in P intersecting E.Since the vertical plane over the real line in C is preserved by PSL2.R/, and

PSL2.C/ acts transitively on the space of all vertical hemispheres (includingplanes), the space of vertical hemispheres in H3 can be identified with the ho-mogeneous space PSL2.R/n PSL2.C/. Since P consists of finitely many A -orbits of circles in C, which corresponds to finitely many A -orbits of pointsin PSL2.R/n PSL2.C/, understanding the asymptotic formula of NP.T;E/ isa special case of the following more general counting problem: letting G DPSL2.C/ and H D PSL2.R/, for a given sequence of growing compact subsetsBT in HnG and a discrete A -orbit v0A in HnG,

what is the asymptotic formula of the number #BT \ v0A ?

If A were of finite co-volume in PSL2.C/, this type of question is well-under-stood due to the works of Duke–Rudnick–Sarnak [15] and Eskin–McMullen[17]. In the next section, we describe analogies/differences of this countingproblem for discrete subgroups of infinite covolume.

3. Counting, mixing, and the Bowen–Margulis–Sullivan measure

Euclidean lattice point counting

We begin with a simple example of the lattice point counting problem in Eu-clidean space. Let G D R3, � D Z3 and let BT WD fx 2 R3 W kxk � T g be theEuclidean ball of radius T centered at the origin. In showing the well-knownfact

#Z3 \ BT � 4�

3T 3;

one way is to count the �-translates of a fundamental domain, say F WDŒ�1

2; 1

2/ Œ�1

2; 1

2/ Œ�1

2; 1

2/ contained in BT , since each translate � C F con-

tains precisely one point, that is, � , from � . We have


Vol.BT �1/

Vol.F /� #f� C F � BT �1 W � 2 Z3g

� #Z3 \ BT � #f� C F � BT C1 W � 2 Z3g � Vol.BT C1/

Vol.F /:

(3.1)

Since Vol.BT ˙1/

Vol.F /D 4�

3.T ˙ 1/3, we obtain that

#Z3 \ BT D 4�

3T 3 CO.T 2/:

This easily generalizes to the following: for any discrete subgroup � in R3

and a sequence BT of compact subsets in R3, we have

#� \ BT D Vol.BT /

Vol.�nR3/CO.Vol.BT /

1��/

provided

� Vol.�nR3/ < 1;� Vol.unit neighborhood of @.BT // D O.Vol.BT /

1��/ for some � > 0.

We have used here that the volume in R3 is computed with respect to theLebesgue measure which is clearly left �-invariant so that it makes sense towrite Vol.�nR3/, and that the ratio Vol.unit neighborhood of @.BT //

Vol.BT /tends to 0 as T !

1.

Hyperbolic lattice point counting

We now consider the hyperbolic lattice counting problem for H3. Let G DPSL2.C/ and � be a torsion-free, cocompact, discrete subgroup ofG. The groupG possesses a Haar measure G which is both left and right invariant underG, in particular, it is left-invariant under � . By abuse of notation, we use thesame notation G for the induced measure on �nG. Fix o D .0; 0; 1/ so thatg 7! g.o/ induces an isomorphism of H3 with G=PSU.2/ and hence G alsoinduces a left G-invariant measure on H3, which will again be denoted by G .Consider the hyperbolic ball BT D fx 2 H3 W d.o; x/ � T g where d is thehyperbolic distance in H3. Then, for a fixed fundamental domain F for � inH3 which contains o in its interior, we have inequalities similar to (3.1):

Vol.BT �d /

Vol.�nG/ � #f�.F / � BT �d W � 2 �g

� #�.o/ \ BT � #f�.F / � BT Cd W � 2 �g � Vol.BT Cd /

Vol.�nG/ ;(3.2)

80 H. Oh

where d is the diameter of F and the volumes Vol.BT ˙d / and Vol.�nG/ arecomputed with respect to G on H3 and �nG respectively.

If we had Vol.BT �d / � Vol.BT Cd / as T ! 1 as in the Euclidean case,we would be able to conclude from here that #�.o/ \ BT � Vol.BT /

Vol.�nG/from

(3.2). However, one can compute that Vol.BT / � c � e2T for some c > 0

and hence the asymptotic formula Vol.BT �d / � Vol.BT Cd / is not true. Thissuggests that the above inequality (3.2) gives too crude estimation of the edgeeffect arising from the intersections of �.F /’s with BT near the boundary ofBT . It turns out that the mixing phenomenon of the geodesic flow on the unittangent bundle T1.�nH3/ with respect to G precisely clears out the fuzzinessof the edge effect. The mixing of the geodesic flow follows from the followingmixing of the frame flow, or equivalently, the decay of matrix coefficients due

to Howe and Moore [28]: Let at WD�et=2 0

0 e�t=2

�.

Theorem 3.3 (Howe–Moore). Let � < G be a lattice. For any 1; 2 2Cc.�nG/,

limt!1

Z�nG

1.gat / 2.g/ dG.g/D 1

G.�nG/Z

�nG

1 dG �Z

�nG

2 dG :

Indeed, using this mixing property of the Haar measure, there are now verywell established counting result due to Duke–Rudnick–Sarnak [15], and Eskin–McMullen [17]: we note that any symmetric subgroup of G is locally isomor-phic to SL2.R/ or SU.2/.

Theorem 3.4 (Duke–Rudnick–Sarnak, Eskin–McMullen). Let H be a sym-metric subgroup of G and � < G a lattice such that H .� \HnH/ < 1, i.e.,H \ � is a lattice in H . Then for any well-rounded sequence BT of compactsubsets in HnG and a discrete �-orbit Œe�� , we have

#Œe�� \ BT � H ..H \ �/nH/G.�nG/ � Vol.BT / as T �! 1.

Here the volume of BT is computed with respect to the invariant measure HnG

on HnG which satisfies G D H ˝ HnG locally.

A sequence fBT � HnGg is called well-rounded with respect to a measure on HnG if the boundaries of BT are -negligible, more precisely, if for allsmall � > 0, the -measure of the �-neighborhood of the boundary of BT isO.� � .BT // as T ! 1.

The idea of using the mixing of the geodesic flow in the counting problemgoes back to Margulis’ 1970 thesis (translated in [34]).

We now consider the case when � < G D PSL2.C/ is not a lattice, that is,G.�nG/ D 1. It turns out that as long as we have a left �-invariant measure,say, on G, satisfying


� .�nG/ < 1;� is the mixing measure for the frame flow on �nG,

then the above heuristics of comparing the counting function for #�.o/\BT tothe volume .BT / can be made into a proof.

For what kind of discrete groups � , do we have a left-�-invariant measureon G satisfying these two conditions? Indeed when � is geometrically finite,the Bowen–Margulis–Sullivan measure mBMS on �nG satisfies these proper-ties. Moreover when � is convex cocompact (that is, geometrically finite withno parabolic elements), the Bowen–Margulis–Sullivan measure is supported ona compact subset of �nG. Therefore � acts cocompactly in the convex hullCH.ƒ.�// of the limit set ƒ.�/; recall that ƒ.�/ is the set of all accumulationpoints of �-orbits on the boundary @.H3/. Hence if we denote by F0 a compactfundamental domain for � in CH.ƒ.�//, the inequality (3.2) continues to holdif we replace the fundamental domain F of � in H3 by F0 and compute thevolumes with respect to mBMS:

QmBMS.BT �d0/

mBMS.�nG/ � #f�.F0/ � BT �d0W � 2 �g

� #�.o/ \ BT � #f�.F0/ � BT Cd0W � 2 �g

� QmBMS.BT Cd0/

mBMS.�nG/ ; (3.5)

where QmBMS is the projection to H3 of the lift of mBMS to G and d0 is thediameter of F0. This suggests a heuristic expectation:

#�.o/ \ BT � QmBMS.BT /

mBMS.�nG/which turns out to be true.

We denote by ı the Hausdorff dimension of ƒ.�/ which is known to beequal to the critical exponent of � . Patterson [46] and Sullivan [56] constructeda unique geometric probability measure o on @.H3/ satisfying that for any� 2 � , ��o is absolutely continuous with respect to o and for any Borelsubset E,

o.�.E// DZ

E

�d.��o/

do

�ıdo:

This measure o is called the Patterson–Sullivan measure viewed from o 2 H3.Then the Bowen–Margulis–Sullivan measure mBMS on T1.H3/ is given by

dmBMS.v/ D f .v/ do.vC/ do.v

�/ dt;where v˙ 2 @.H3/ are the forward and the backward endpoints of the geodesicdetermined by v and t D ˇv�.o; v/ measures the signed distance of the horo-spheres based at v� passing through o and v. The density function f is givenby f .v/ D eı.ˇ

vC .o;v/Cˇv� .o;v// so that mBMS is left �-invariant. Clearly, the

82 H. Oh

support of mBMS is given by the set of v with v˙ � ƒ.�/. Noting that T1.H3/

is isomorphic to G=M where M D fdiag.ei� ; e�i� /g, we will extend mBMS toan M -invariant measure on G. We use the same notation mBMS to denote themeasure induced on �nG.

Theorem 3.6. For � geometrically finite and Zariski dense,

(1) Finiteness: mBMS.�nG/ < 1;(2) Mixing: For any 1; 2 2 Cc.�nG/, as t ! 1,Z

�nG

1.gat / 2.g/ dmBMS.g/ �! 1

mBMS.�nG/Z

�nG

1 dmBMS

Z�nG

2 dmBMS:

The finiteness result (1) is due to Sullivan [56] and the mixing result (2) forframe flow is due to Flaminio–Spatzier [18] and Winter [59] based on the workof Rudolph [50] and Babillot [1].

In order to state an analogue of Theorem 3.4 for a general geometricallyfinite group, we need to impose a condition on .H \ �/nH analogous to thefiniteness of the volume H .� \HnH/. In [45], we define the so called skin-ning measure PS

H on .� \H/nH , which is intuitively the slice measure on Hof mBMS. We note that PS

H depends on � , not only on H \ � . A finiteness cri-terion for PS

H is given in [45]. The following is obtained in [45] non-effectivelyand [39] effectively.

Theorem 3.7 (O.–Shah, Mohammadi–O.). LetH be a symmetric subgroup ofG and � < G a geometrically finite and Zariski dense subgroup. Suppose thatthe skinning measure of H \ �nH is finite, i.e., PS

H .� \ HnH/ < 1. Thenthere exists an explicit locally finite Borel measure MHnG on HnG such thatfor any well-rounded sequence BT of compact subsets in HnG with respect toMHnG and a discrete �-orbit Œe�� , we have

#Œe�� \ BT � PSH ..H \ �/nH/mBMS.�nG/ � MHnG.BT / as T �! 1:

A special case of this theorem implies Theorem 1.8, modulo the computationof the measure MHnG.ET / where ET is given in (2.2). We mention that in thecase when the critical exponent ı of � is strictly bigger than 1, both Theorem 3.7and Theorem 1.8 can be effectivized by [39].

The reason that we have the ˛-dimensional Hausdorff measure in the state-ment of Theorem 1.8 is because the slice measure of mBMS on each horizon-tal plane is the Patterson–Sullivan measure multiplied with a correct densityfunction needed for the �-invariance, which turns out to coincide with the ı-dimensional Hausdorff measure on the limit set of ƒ.�/ when all cusps of �are of rank at most 1, which is the case for the Apollonian group.

Counting problems for �-orbits in HnG are technically much more in-volved when H is non-compact than when H is compact, and relies on un-derstanding the asymptotic distribution of �n�Hat in �nG as t ! 1. When


H D PSL2.R/, the translate �n�Hat corresponds to the orthogonal translateof a totally geodesic surface for time t , and we showed that, after the correctscaling of e.2�ı/t , �n�Hat becomes equidistributed in �nG with respect tothe Burger–Roblin measure mBR, which is the unique non-trivial ergodic horo-spherical invariant measure on �nG. We refer to [42], [43], [44] for more de-tails.

More circle packings

This viewpoint of approaching Apollonian circle packings via the study ofKleinian groups allows us to deal with more general circle packings, providedthey are invariant under a non-elementary geometrically finite Kleinian group.

One way to construct such circle packings is as follows:

Example 3.8. Let X be a finite volume hyperbolic 3-manifold with non-emptytotally geodesic boundary. Then

� � WD �1.X/ is a geometrically finite Kleinian group;� By developing X in the upper half space H3, the domain of discontinuity�.�/ WD OC�ƒ.�/ consists of the disjoint union of open disks (correspond-ing to the boundary components of the universal cover eX).

Set P to be the union of circles which are boundaries of the disks in �.�/.In this case, Res.P/ defined as the closure of all circles in P is equal to thelimit set ƒ.�/.

In Sect. 2, we explained how Apollonian circle packings can be described inthis way.

Figure 7, due to McMullen, is also an example of a circle packing obtainedin this way, here the symmetry group � is the fundamental group of a compact

Fig. 7. Sierpinski curve

84 H. Oh

hyperbolic 3-manifold with totally geodesic boundary being a compact surfaceof genus two. This limit set is called a Sierpinski curve, being homeomorphic tothe well-known Sierpinski Carpet.

Many more pictures of circle packings constructed in this way can be foundin the book “Indra’s pearls” by Mumford, Series and Wright (Cambridge Univ.Press 2002).

For P constructed in Example 3.8, we define as before NP.T;E/ WD #fC2 P W C \E ¤ ;; curv.C / � T g for any bounded Borel subset E in C.

Theorem 3.9 (O.–Shah, [45]). There exist a constant c� > 0 and a locallyfinite Borel measure !P on Res.P/ such that for any bounded Borel subsetE � C with [email protected]// D 0,

NP.T;E/ � c� � !P.E/ � T ı as T �! 1;

where ı D dimH .Res.P//. Moreover, if � is convex cocompact or if the cuspsof � have rank at most 1, then !P coincides with the ı-dimensional Hausdorffmeasure on Res.P/.

We refer to [45] for the statement for more general circle packings.

4. Integral Apollonian circle packings

We call an Apollonian circle packing P integral if every circle in P has inte-gral curvature. Does there exist any integral P? The answer is positive thanksto the following beautiful theorem of Descartes:


Theorem 4.1 (Descartes 1643, [14]). A quadruple .a; b; c; d/ is the curvaturesof four mutually tangent circles if and only if it satisfies the quadratic equation:

2.a2 C b2 C c2 C d2/ D .aC b C c C d/2:

In the above theorem, we ask circles to be oriented so that their interiors aredisjoint with each other. For instance, according to this rule, the quadruple ofcurvatures of four largest four circles in Fig. 3 is .�1; 2; 2; 3/ or .1;�2;�2;�3/,for which we can easily check the validity of the Descartes theorem: 2..�1/2 C22 C 22 C 32/ D 36 D .�1C 2C 2C 3/2.

In what follows, we will always assign the negative curvature to the largestbounding circle in a bounded Apollonian packing, so that all other circles willthen have positive curvatures.

Given three mutually tangent circles of curvatures a; b; c, the curvatures, say,d and d 0, of the two circles tangent to all three must satisfy 2.a2 C b2 C c2 Cd2/ D .a C b C c C d/2 and 2.a2 C b2 C c2 C .d 0/2/ D .a C b C c C d 0/2by the Descartes theorem. By subtracting the first equation from the second, weobtain the linear equation:

d C d 0 D 2.aC b C c/:

So, if a; b; c; d are integers, so is d 0. Since the curvature of every circle from thesecond generation or later is d 0 for some 4mutually tangent circles of curvaturesa; b; c; d from the previous generation, we deduce:

Theorem 4.2 (Soddy 1937). If the initial 4 circles in an Apollonian packing Phave integral curvatures, P is integral.

Combined with Descartes’ theorem, for any integral solution of 2.a2 Cb2 Cc2 C d2/ D .a C b C c C d/2, there exists an integral Apollonian packing!Because the smallest positive curvature must be at least 1, an integral Apollonianpacking cannot have arbitrarily large circles. In fact, any integral Apollonianpacking is either bounded or lies between two parallel lines.

Fig. 8. Integral Apollonian packings

For a given integral Apollonian packing P , it is natural to inquire about itsDiophantine properties such as

86 H. Oh

Question 4.3. � Are there infinitely many circles with prime curvatures?� Which integers appear as curvatures?

We call P primitive, if g:c:d:C 2P.curv.C // D 1. We call a circle is primeif its curvature is a prime number, and a pair of tangent prime circles will becalled twin prime circles. There are no triplet primes of three mutually tangentcircles, all having odd prime curvatures.

Theorem 4.4 (Sarnak 2007). There are infinitely many prime circles as well astwin prime circles in any primitive integral Apollonian packing.

In the rest of this section, we let P be a bounded primitive integral Apol-lonian packing. Theorem 4.4 can be viewed as an analogue of the infinitude ofprime numbers. In order to formulate what can be considered as an analogue ofthe prime number theorem, we set

…T .P/ WD #fprime C 2 P W curv.C / � T gand

….2/T .P/ WD #ftwin primes C1; C2 2 P W curv.Ci / � T g:

Using the sieve method based on heuristics on the randomness of Möbiusfunction, Fuchs and Sanden [21] conjectured:

Conjecture 4.5 (Fuchs–Sanden).

…T .P/ � c1NP.T /

logTI …

.2/T .P/ � c2

NP.T /

.logT /2;

where c1 > 0 and c2 > 0 can be given explicitly.

Based on the breakthrough of Bourgain, Gamburd, and Sarnak [5] provingthat the Cayley graphs of congruence quotients of the integral Apollonian groupform an expander family, together with Selberg’s upper bound sieve, we obtainupper bounds of true order of magnitude:

Theorem 4.6 (Kontorovich–O. [30]). For T 1,

� …T .P/ � T ˛

log T;

� ….2/T .P/ � T ˛

.log T /2 .

The lower bounds for Conjecture 4.5 are still open and very challenging.However a problem which is more amenable to current technology is to countcurvatures without multiplicity. Our counting Theorem 1.5 for circles says thatthe number of integers at most T arising as curvatures of circles in integral Pcounted with multiplicity, is of order T 1:3:::. So one may hope that a positivedensity (Dproportion) of integers arises as curvatures, as conjectured by Gra-ham, Lagarias, Mallows, Wilkes, and Yan (Positive density conjecture) [23].


Theorem 4.7 (Bourgain–Fuchs [4]). For a primitive integral Apollonian pack-ing P ,

#fcurv.C / � T W C 2 Pg T:

A stronger conjecture, called the Strong Density conjecture, of Graham et al.says that every integer occurs as the value of a curvature of a circle in P , unlessthere are congruence obstructions. Fuchs [19] showed that the only congruenceobstructions are modulo 24, and hence the strong positive density conjecture (orthe local-global principle conjecture) says that every sufficiently large integerwhich is congruent to a curvature of a circle in P modulo 24 must occur as thevalue of a curvature of some circle in P . This conjecture is still open, but thereis now a stronger version of the positive density theorem:

Theorem 4.8 (Bourgain–Kontorovich [7]). For a primitive integral Apollo-nian packing P ,

#fcurv.C / � T W C 2 Pg � �.P/

24� T;

where �.P/ > 0 is the number of residue classes modulo 24 of curvatures ofP .

Improving Sarnak’s result on the infinitude of prime circles, Bourgainshowed that a positive fraction of prime numbers appear as curvatures in P .

Theorem 4.9 (Bourgain [3]).

#fprime curv.C / � T W C 2 Pg T

logT:

Integral Apollonian group

In studying the Diophantine properties of integral Apollonian packings, wework with the integral Apollonian group, rather than the geometric Apolloniangroup which was defined in Sect. 2.

We call a quadruple .a; b; c; d/ a Descartes quadruple if it represents curva-tures of four mutually tangent circles (oriented so that their interiors are disjoint)in the plane. By Descartes’ theorem, any Descartes quadruple .a; b; c; d/ lies onthe cone Q.x/ D 0, where Q denotes the so-called Descartes quadratic form

Q.a; b; c; d/ D 2.a2 C b2 C c2 C d2/ � .aC b C c C d/2:

The quadratic form Q has signature .3; 1/ and hence over the reals, the or-thogonal group OQ is isomorphic to O.3; 1/, which is the isometry group of thehyperbolic 3-space H3.

88 H. Oh

We observe that if .a; b; c; d/ and .a; b; c; d 0/ are Descartes quadruples, thend 0 D �d C 2.aC b C c/ and hence .a; b; c; d 0/ D .a; b; c; d/S4 where

S1 D

0B@

�1 0 0 02 1 0 0

2 0 1 0

2 0 0 1

1CA ; S2 D

0B@1 2 0 0

0 �1 0 00 2 1 0

0 2 0 1

1CA ;

S3 D

0B@1 0 2 0

0 1 2 0

0 0 �1 00 0 2 1

1CA ; S4 D

0B@1 0 0 2

0 1 0 2

0 0 1 2

0 0 0 �1

1CA :

Now the integral Apollonian group A is generated by those four reflectionsS1; S2; S3; S4 in GL4.Z/ and one can check that A < OQ.Z/.

Fixing an integral Apollonian circle packing P , all Descartes quadruplesassociated to P is a single A -orbit in the cone Q D 0. Moreover if wechoose a root quadruple vP from P , which consists of curvatures of fourlargest mutually tangent circles, any reduced word wn D vPSi1

� � �Sinwith

Sij 2 fS1; S2; S3; S4g is obtained from wn�1 D vPSi1� � �Sin�1

by changingprecisely one entry and this new entry is the maximum entry of wn, which is thecurvature of a precisely one new circle added at the n-th generation [23]. Thisgives us the translation of the circle counting problem for a bounded Apollonianpackings as the orbital counting problem of an A -orbit in a cone Q D 0:

NT .P/ D #fv 2 vPA W kvkmax � T g C 3:

The integral Apollonian group A is isomorphic to the geometric Apolloniangroup AP (the subgroup generated by four inversions with respect to the dualcircles of four mutually tangent circles in P): there exists an explicit isomor-phism between the orthogonal group OQ and Möb. OC/ which maps the integralApollonian group A to the geometric Apollonian group AP . In particular, Ais a subgroup OQ.Z/ which is of infinite index and Zariski dense in OQ. Sucha subgroup is called a thin group. Diophantine properties of an integral Apollo-nian packing is now reduced to the study of Diophantine properties of an orbitof the thin group A . Unlike orbits under an arithmetic subgroup (subgroups ofOQ.Z/ of finite index) which has a rich theory of automorphic forms and er-godic theory, the study of thin groups has begun very recently, but with a greatsuccess. In particular, the recent developments in expanders is one of key ingre-dients in studying primes or almost primes in thin orbits (see [3]).

5. Expanders and sieve

All graphs will be assumed to be simple (no multiple edges and no loops) andconnected in this section. For a finite k-regular graph X D X.V;E/ with V Dfv1; : : : ; vng the set of vertices and E the set of edges, the adjacency matrix


A D .aij / is defined by aij D 1 if fvi ; vj g 2 E and aij D 0 otherwise. Since Ais a symmetric real matrix, it has n real eigenvalues: 0.X/ � 1.X/ � � � � � n�1.X/. As X is simple and connected, the largest eigenvalue 0.X/ is givenby k and has multiplicity one.

Definition 5.1. A family of k-regular graphs fXig with (#Xi ! 1) is called anexpander family if there exists an �0 > 0 such that

supi

1.Xi / � k � �0:

Equivalently, fXig is an expander family if there exists a uniform positivelower bound for the Cheeger constant (or isoperimetric constant)

h.Xi / WD min0<#W �#Xi =2

#@.W /#W

;

where @.W / means the set of edges with exactly one vertex in W . Note thatthe bigger the Cheeger constant is, the harder it is to break the graph into twopieces. Intuitively speaking, an expander family is a family of sparse graphs (asthe regularity k is fixed) with high connectivity properties (uniform lower boundfor the Cheeger constants).

Although it was known that there has to be many expander families usingprobabilistic arguments due to Pinsker, the first explicit construction of an ex-pander family is due to Margulis in 1973 [33] using the representation theoryof a simple algebraic group and automorphic form theory. We explain his con-struction below; strictly speaking, what we describe below is not exactly sameas his original construction but the idea of using the representation theory of anambient algebraic group is the main point of his construction as well as in theexamples below.

Let G be a connected simple non-compact real algebraic group defined overQ, with a fixed Q-embedding into SLN . Let G.Z/ WD G \ SLN .Z/ and � <

G.Z/ be a finitely generated subgroup. For each positive integer q, the principalcongruence subgroup �.q/ of level q is defined to be f� 2 � W � D e mod qg.

Fix a finite symmetric generating subset S for � . Then S generates the group�.q/n� via the canonical projection. We denote by Xq WD C .�.q/n�; S/ theCayley graph of the group �.q/n� with respect to S , that is, vertices of Xq areelements of �.q/n� and two elements g1; g2 form an edge if g1 D g2s forsome s 2 S . Then Xq is a connected k-regular graph for k D #S . Now a keyobservation due to Margulis is that if � is of finite index inG.Z/, or equivalentlyif � is a lattice in G, then the following two properties are equivalent: for anyI � N,

(1) The family fXq W q 2 I g is an expander;(2) The trivial representation 1G is isolated in the sum

Lq2I L

2.�.q/nG/ inthe Fell topology of the set of unitary representations of G.

90 H. Oh

We won’t give a precise definition of the Fell topology, but just say that thesecond property is equivalent to the following: for a fixed compact generatingsubsetQ ofG, there exists � > 0 (independent of q 2 I ) such that any unit vec-tor f 2 L2.�.q/nG/ satisfying maxq2Q kq:f � f k < � is G-invariant, i.e., aconstant. Briefly speaking, it follows almost immediately from the definition ofan expander family that the family fXqg is an expander if and only if the triv-ial representation 1� of � is isolated in the sum

Lq L

2.�.q/n�/. On the otherhand, the induced representation of 1� from � toG isL2.�nG/, which containsthe trivial representation 1G , if � is a lattice in G. Therefore, by the continuityof the induction process, the weak-containment of 1� in

Lq L

2.�.q/n�/ im-plies the weak-containment of 1G in

Lq L

2.�.q/nG/, which explains why (2)implies (1).

The isolation property of 1G as in (2) holds for G; if the real rank of G is atleast 2 or G is a rank one group of type Sp.m; 1/ or F�20

4 , G has the so-calledKazhdan’s property (T) [29], which says that the trivial representation of G isisolated in the whole unitary dual of G. When G is isomorphic to SO.m; 1/or SU.m; 1/ which do not have Kazhdan’s property (T), the isolation of thetrivial representation is still true in the subset of all automorphic representationsL2.�.q/nG/’s, due to the work of Selberg, Burger–Sarnak [11] and Clozel [13].This latter property is referred as the phenomenon that G has property � withrespect to the congruence family f�.q/g.

Therefore, we have:

Theorem 5.2. If � is of finite index in G.Z/, then the family fXq D C .�.q/n�;S/ W q 2 Ng is an expander family.

In the case when � is of infinite index, the trivial representation is not con-tained in L2.�.q/nG/, as the constant function is not square-integrable, and theabove correspondence cannot be used, and deciding whether Xq forms an ex-pander or not for a thin group was a longstanding open problem. For instance,

if Sk consists of four matrices�1 ˙k0 1

�and

�1 0

˙k 1�

, then the group �k gen-

erated by Sk has finite index only for k D 1; 2 and hence we know the familyfXq.k/ D C .�k.q/n�k; Sk/g forms an expander for k D 1; 2 by Theorem 5.2but the subgroup �3 generated by S3 has infinite index in SL2.Z/ and it wasnot known whether fXq.3/g is an expander family until the work of Bourgain,Gamburd and Sarnak [5].

Theorem 5.3 (Bourgain–Gamburd–Sarnak [5], Salehi–Golsefidy–Varjú[51]). Let � < G.Z/ be a thin subgroup, i.e., � is Zariski dense inG. Let S be afinite symmetric generating subset of � . Then fC .�.q/n�; S/ W q: square-freegforms an expander family.


If G is simply connected in addition, the strong approximation theorem ofMatthews, Vaserstein and Weisfeiler [35] says that there is a finite set B ofprimes such that for all q with no prime factors from B, �.q/n� is isomor-phic to the finite group G.Z=qZ/ via the canonical projection � ! G.Z=qZ/;hence the corresponding Cayley graph C .G.Z=qZ/; S/ is connected. Similarly,Theorem 5.3 says that the Cayley graphs C .G.Z=qZ/; S/ with q square-freeand with no factors from B are highly connected, forming an expander family;called the super-strong approximation theorem.

The proof of Theorem 5.3 is based on additive combinatorics and Helf-gott’s work on approximate subgroups [27] and generalizations made by Pyber–Szabó [47] and Breuillard–Green–Tao [9] (see also [25]).

The study of expanders has many surprising applications in various areas ofmathematics (see [54]). We describe its application in sieves, i.e., in the studyof primes. For motivation, we begin by considering an integral polynomial f 2ZŒx�. The following is a basic question:

Are there infinitely many integers n 2 Z such that f .n/ is prime?

� If f .x/ D x, the answer is yes; this is the infinitude of primes.� If f .x/ D ax C b, the answer is yes if and only if a; b are co-prime. This is

Dirichlet’s theorem.� If f .x/ D x.xC2/, then there are no primes in f .Z/ for an obvious reason.

On the other hand, twin prime conjecture says that there are infinitely manyn’s such that f .n/ is a product of at most 2 primes. Indeed, Brun introducedwhat is called Brun’s combinatorial sieve to attack this type of question,and proved that there are infinitely many n’s such that f .n/ D n.n C 2/

is 20-almost prime, i.e., a product of at most 20 primes. His approach wasimproved by Chen [12] who was able to show such a tantalizing theorem thatthere are infinitely many n’s such that f .n/ D n.nC 2/ is 3-almost prime.

In view of the last example, the correct question is formulated as follows:Is there R < 1 such that the set of n 2 Z such that f .n/ is R-almost

prime is infinite?Bourgain, Gamburd and Sarnak [6] made a beautiful observation that Brun’s

combinatorial sieve can also be implemented for orbits of � on an affine spacevia affine linear transformations and the expander property of the Cayley graphsof the congruence quotients of � provides a crucial input needed in executingthe sieve machine.

Continuing our setup that G � SLN and � < G.Z/, we consider the orbitO D v0� � ZN for a non-zero v0 2 ZN and let f 2 QŒx1; : : : ; xN � such thatf .O/ � Z.

Theorem 5.4 (Bourgain–Gamburd–Sarnak[6], Sarnak–Salehi–Golsefidy[52]). There exists R D R.O; f / � 1 such that the set of vectors v 2 Osuch that f .v/ is R-almost prime is Zariski dense in v0G.

92 H. Oh

We ask the following finer question:

Describe the distribution of the set fv 2 O W f .v/ is R-almost primegin the variety v0G.

In other words, is the set in concern focused in certain directions in O or equi-distributed in O? This is a very challenging question at least in the same gener-ality as the above theorem, but when G is the orthogonal group of the Descartesquadratic form Q, Q.v0/ D 0, and � is the integral Apollonian group, we areable to give more or less a satisfactory answer by [30] and [32]. More generally,we have the following: Let F be an integral quadratic form of signature .n; 1/and let � < SOF .Z/ be a geometrically finite Zariski dense subgroup. Supposethat the critical exponent ı of � is bigger than .n � 1/=2 if n D 2; 3 and biggerthan n�2 if n � 4. Let v0 2 ZnC1 be non-zero and O WD v0� . We also assumethat the skinning measure associated to v0 and � is finite.

Theorem 5.5 (Mohammadi–O. [39]). Let f D f1 � � � fk 2 QŒx1; : : : ; xnC1�

be a polynomial with each fi absolutely irreducible and distinct with rationalcoefficients and fi .O/ � Z. Then we construct an explicit locally finite measureM on the variety v0G, depending on � such that for any family BT of subsetsin v0G which is effectively well-rounded with respect to M , we have

(1) Upper bound: #fv 2 O \ BT W each fi .v/ is primeg � M .BT /

.log M .BT //k ;

(2) Lower bound: Assuming further that maxx2BTkxk � M .BT /

ˇ for someˇ > 0, there exists R D R.O; f / > 1 such that

#fv 2 O \ BT W f .v/ is R-almost primeg M .BT /

.log M .BT //k:

The terminology of BT being effectively well-rounded with respect to Mmeans that there exists p > 0 such that for all small � > 0 and for all T 1,the M -measure of the �-neighborhood of the boundary of BT is at most oforder O.�pM .BT // with the implied constant independent of � and T . Forinstance, the norm balls fv 2 v0G W kvk � T g and many sectors are effectivelywell-rounded (cf. [39]).

When � is of finite index, M is just a G-invariant measure on v0G and thistheorem was proved earlier by Nevo–Sarnak [41] and Gorodnik–Nevo [22].

If Q is the Descartes quadratic form, A is the integral Apollonian group,and BT D fv 2 R4 W Q.v/ D 0; kvkmax � T g is the max-norm ball, then forany primitive integral Apollonian packing P , the number of prime circles in Pof curvature at most T is bounded by

4XiD1

#fv 2 vPA \ BT ; f .v/ WD vi primeg


which is bounded by M .BT /log.M .BT //

by Theorem 5.5. Since we have M .BT / Dc �T ˛ CO.T ˛��/ where ˛ D 1:305 : : : is the critical exponent of A , this givesan upper bound T ˛= logT for the number of prime circles of curvature at mostT , as stated in Theorem 4.6. The upper bound for twin prime circle count canbe done similarly with f .v/ D vivj .

Here are a few words on Brun’s combinatorial sieve and its use in Theo-rem 5.5. Let A D famg be a sequence of non-negative numbers and let B bea finite set of primes. For z > 1, let Pz D Q

p…B;p<z p and S.A; Pz/ WDP.m;Pz/D1 am. To estimate Sz WD S.A; Pz/, we need to understand how A is

distributed along arithmetic progressions. For q square-free, define

Aq WD fam 2 A W n � 0.q/gand set jAqj WD P

m�0.q/ am.We use the following combinatorial sieve (see [26, Theorem 7.4]):

Theorem 5.6. .A1/ For q square-free with no factors in B , suppose that

jAqj D g.q/X C rq.A/;

where g is a function on square-free integers with 0 � g.p/ < 1, g is mul-tiplicative outside B , i.e., g.d1d2/ D g.d1/g.d2/ if d1 and d2 are square-free integers with .d1; d2/ D 1 and .d1d2; B/ D 1, and for some c1 > 0,g.p/ < 1 � 1=c1 for all prime p … B .

.A2/ A has level distribution D, in the sense that for some � > 0 and C� > 0,Xq<D

jrq.A/j � C�X1��:

.A3/ A has sieve dimension k in the sense that there exists c2 > 0 such that forall 2 � w � z,

�c2 �X

.p;B/D1;w�p�z

g.p/ logp � r logz

w� c2:

Then for s > 9r , z D D1=s and X large enough,

S.A; Pz/ X

.log X /k:

For our orbit O D v0� and f as in Theorem 5.5, we set

am.T / WD #fx 2 O \ BT W f .x/ D mgI�v0

.q/ WD f� 2 � W v0� � v0 .q/g;jA.T /j WD

Xm

am.T / D #O \ BT I

jAq.T /j WDX

m�0.q/

am.T / D #fx 2 O \ BT W f .x/ � 0 .q/g:

94 H. Oh

Suppose we can verify the sieve axioms for these sequences Aq.T / and z oforder T �. Observe that if .f .v/ D m;Pz/ D 1, then all prime factors ofm haveto be at least of order z D T �. It follows that if f .v/ D m has R prime factors,then T �R � m � T degree.f /, and hence R � .degree.f //=�. Therefore,Sz WD P

.m;Pz/D1 am.T / gives an estimate of the number of all v 2 O suchthat f .v/ is R-almost prime for R D .degree.f //=�.

In order to verify these sieve axioms for O D v0� , we replace � by itspreimage under the spin cover eG of G, so that � satisfies the strong approxi-mation property that �.q/n� D eG.Z=qZ/ outside a fixed finite set of primes.The most crucial condition is to understand the distribution of am.T /’s alongthe arithmetic progressions, i.e.,

PmD0 .q/ am.T / for all square-free integers q,

more precisely, we need to have a uniform control on the remainder term rq ofAq.T / D P

mD0 .q/ am D g.q/X C rq such as rq � X 1�� for some � > 0

independent of q. By writing

Aq.T / DX

�2�v0.q/n�;f .v0�/D0 .q/

#.v0�v0.q/� \ BT /

the following uniform counting estimates provide such a control on the remain-der term:

Theorem 5.7 (Mohammadi–O. [39]). Let � and BT be as in Theorem 5.5. Forany � 2 � and any square-free integer q,

#v0�.q/� \ BT D c0

Œ� W �.q/�M .BT /CO.M .BT /1��/;

where c0 > 0 and � > 0 are independent over all � 2 � and q.

A basic ingredient of Theorem 5.7 is a uniform spectral gap for the Laplacianacting on L2.�.q/nHn/. Note that zero is no more the base-eigenvalue of theLaplacian when �.q/ is a thin group, but ı.n�1�ı/ is by Sullivan [57] and Lax–Phillips [31]. However, the expander result (Theorem 5.3) implies a uniformlower bound for the gap between the base eigenvalue ı.n � 1 � ı/ and the nextone; this transfer property was obtained by Bourgain, Gamburd and Sarnak. Asexplained in Sect. 3, the mixing of frame flow of the Bowen–Margulis–Sullivanmeasure is a crucial ingredient in obtaining the main term in Theorem 5.7, andthe (uniform) error term in the counting statement of Theorem 5.7 is again aconsequence of a uniform error term in the effective mixing of frame flow, atleast under our hypothesis on ı.

References

[1] M. Babillot, On the mixing property for hyperbolic systems, Israel J. Math., 129 (2002),61–76.

[2] J. Bourgain, Integral Apollonian circle packings and prime curvatures, preprint,arXiv:1105.5127.


[3] J. Bourgain, Some Diophantine applications of the theory of group expansion, In: ThinGroups and Superstrong Approximation, (eds. E. Breulliard and H. Oh), Math. Sci. Res.Inst. Publ., 61, Cambridge Univ. Press, Cambridge, to appear (2014).

[4] J. Bourgain and E. Fuchs, A proof of the positive density conjecture for integer Apollonianpackings, J. Amer. Math. Soc., 24 (2011), 945–967.

[5] J. Bourgain, A. Gamburd and P. Sarnak, Affine linear sieve, expanders, and sum-product,Invent. Math., 179 (2010), 559–644.

[6] J. Bourgain, A. Gamburd and P. Sarnak, Generalization of Selberg’s 3=16 theorem andaffine sieve, Acta Math., 207 (2011), 255–290.

[7] J. Bourgain and A. Kontorovich, On the local-global conjecture for integral Apolloniangaskets, preprint, arXiv:1205.4416; Invent. Math., to appear.

[8] D.W. Boyd, The sequence of radii of the Apollonian packing, Math. Comp., 39 (1982),249–254.

[9] E. Breuillard, B. Green and T. Tao, Approximate subgroups of linear groups, Geom. Funct.Anal., 21 (2011), 774–819.

[10] M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779–803.

[11] M. Burger and P. Sarnak, Ramanujan duals. II, Invent. Math., 106 (1991), 1–11.[12] J.R. Chen, On the representation of a larger even integer as the sum of a prime and the

product of at most two primes, Sci. Sinica, 16 (1973), 157–176.[13] L. Clozel, Démonstration de la conjecture � , Invent. Math., 151 (2003), 297–328.[14] H.S.M. Coxeter, The problem of Apollonius, Amer. Math. Monthly, 75 (1968), 5–15.[15] W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous vari-

eties, Duke Math. J., 71 (1993), 143–179.[16] N. Eriksson and J.C. Lagarias, Apollonian circle packings: number theory. II. Spherical and

hyperbolic packings, Ramanujan J., 14 (2007), 437–469.[17] A. Eskin and C.T. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke

Math. J., 71 (1993), 181–209.[18] L. Flaminio and R.J. Spatzier, Geometrically finite groups, Patterson–Sullivan measures

and Ratner’s theorem, Invent. Math., 99 (1990), 601–626.[19] E. Fuchs, Strong approximation in the Apollonian group, J. Number Theory, 131 (2011),

2282–2302.[20] E. Fuchs, Counting problems in Apollonian packings, Bull. Amer. Math. Soc. (N.S.), 50

(2013), 229–266.[21] E. Fuchs and K. Sanden, Some experiments with integral Apollonian circle packings, Exp.

Math., 20 (2011), 380–399.[22] A. Gorodnik and A. Nevo, Lifting, restricting and sifting integral points on affine homoge-

neous varieties, Compos. Math., 148 (2012), 1695–1716.[23] R.L. Graham, J.C. Lagarias, C.L. Mallows, A.R. Wilks and C.H. Yan, Apollonian circle

packings: number theory, J. Number Theory, 100 (2003), 1–45.[24] R.L. Graham, J.C. Lagarias, C.L. Mallows, A.R. Wilks and C.H. Yan, Apollonian circle

packings: geometry and group theory. I. The Apollonian group, Discrete Comput. Geom.,34 (2005), 547–585.

[25] B. Green, Approximate groups and their applications: work of Bourgain, Gamburd, Helfgottand Sarnak, Current Events Bulletin of the AMS, 2010, preprint, arXiv:0911.3354.

[26] H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 1974.[27] H.A. Helfgott, Growth and generation in SL2.Z=pZ/, Ann. of Math. (2), 167 (2008), 601–

623.[28] R.E. Howe and C.C. Moore, Asymptotic properties of unitary representations, J. Funct.

Anal., 32 (1979), 72–96.

96 H. Oh

[29] D.A. Kazhdan, Connection of the dual space of a group with the structure of its closesubgroups, Funct. Anal. Appl., 1 (1967), 63–65.

[30] A. Kontorovich and H. Oh, Apollonian circle packings and closed horospheres on hyper-bolic 3-manifolds, J. Amer. Math. Soc., 24 (2011), 603–648.

[31] P.D. Lax and R.S. Phillips, The asymptotic distribution of lattice points in Euclidean andnon-Euclidean spaces, J. Funct. Anal., 46 (1982), 280–350.

[32] M. Lee and H. Oh, Effective circle count for Apollonian packings and closed horospheres,Geom. Funct. Anal., 23 (2013), 580–621.

[33] G.A. Margulis, Explicit constructions of expanders, Problems Inform. Transmission, 9(1973), 325–332.

[34] G.A. Margulis, On Some Aspects of the Theory of Anosov Systems. With a surveyby Richard Sharp: Periodic orbits of hyperbolic flows. Translated from the Russian byValentina Vladimirovna Szulikowska, Springer Monogr. Math., Springer-Verlag, 2004.

[35] C.R. Matthews, L.N. Vaserstein and B. Weisfeiler, Congruence properties of Zariski-densesubgroups. I, Proc. London Math. Soc. (3), 48 (1984), 514–532.

[36] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Stud. Adv.Math., 44, Cambridge Univ. Press, Cambridge, 1995.

[37] R.D. Mauldin and M. Urbanski, Dimension and measures for a curvilinear Sierpinski gasketor Apollonian packings, Adv. Math., 136 (1998), 26–38.

[38] C.T. McMullen, Hausdorff dimension and conformal dynamics. III. Computation of dimen-sion, Amer. J. Math., 120 (1998), 691–721.

[39] A. Mohammadi and H. Oh, Matrix coefficients, counting and primes for geometrically finitegroups, preprint, arXiv:1208.4139; J. Eur. Math. Soc. (JEMS), to appear.

[40] D. Mumford, C. Series and D. Wright, Indra’s Pearls, Cambridge Univ. Press, New York,2002.

[41] A. Nevo and P. Sarnak, Prime and almost prime integral points on principal homogeneousspaces, Acta Math., 205 (2010), 361–402.

[42] H. Oh, Dynamics on geometrically finite hyperbolic manifolds with applications to Apol-lonian circle packings and beyond, In: Proc. of International Congress of Mathematicians,Hyderabad, 2010, Vol. III, Hindustan Book Agency, New Delhi, 2010, pp. 1308–1331.

[43] H. Oh, Harmonic analysis, ergodic theory and counting for thin groups, In: Thin Groupsand Superstrong Approximation, (eds. E. Breulliard and H. Oh), Math. Sci. Res. Inst. Publ.,61, Cambridge Univ. Press, Cambridge, to appear (2014).

[44] H. Oh and N.A. Shah, The asymptotic distribution of circles in the orbits of Kleinian groups,Invent. Math., 187 (2012), 1–35.

[45] H. Oh and N.A. Shah, Equidistribution and counting for orbits of geometrically finite hy-perbolic groups, J. Amer. Math. Soc., 26 (2013), 511–562.

[46] S.J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241–273.[47] L. Pyber and E. Szabó, Growth in finite simple groups of Lie type of bounded rank, preprint,

arXiv:1005.1858.[48] J.G. Ratcliffe, Foundations of Hyperbolic Manifolds, Grad. Texts in Math., 149, Springer-

Verlag, 1994.[49] T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.),

95, Math. Soc. France, 2003.[50] D. Rudolph, Ergodic behaviour of Sullivan’s geometric measure on a geometrically finite

hyperbolic manifold, Ergodic Theory Dynam. Systems, 2 (1982), 491–512.[51] A. Salehi Golsefidy and P. Sarnak, The affine sieve, J. Amer. Math. Soc., 26 (2013), 1085–

1105.[52] A. Salehi Golsefidy and P. Varjú, Expansion in perfect groups, Geom. Funct. Anal., 22

(2012), 1832–1891.


[53] P. Sarnak, Integral Apollonian packings, Amer. Math. Monthly, 118 (2011), 291–306.[54] P. Sarnak, Growth in linear groups, In: Thin Groups and Superstrong Approximation, (eds.

E. Breulliard and H. Oh), Math. Sci. Res. Inst. Publ., 61, Cambridge Univ. Press, Cam-bridge, to appear (2014).

[55] B. Stratmann and M. Urbanski, The box-counting dimension for geometrically finiteKleinian groups, Fund. Math., 149 (1996), 83–93.

[56] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. HautesÉtudes Sci. Publ. Math., 50 (1979), 171–202.

[57] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finiteKleinian groups, Acta Math., 153 (1984), 259–277.

[58] I. Vinogradov, Effective bisector estimate with applications to Apollonian circle packings,preprint, arXiv:1204.5498.

[59] D. Winter, Mixing of frame flow for rank one locally symmetric spaces and measure clas-sification, preprint, 2013.

apollonian circle packings: dynamics and number theory

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