APOLLONIAN CIRCLE PACKINGS: DYNAMICS AND …apollonian circle packings: dynamics and number ... 121...
27
APOLLONIAN CIRCLE PACKINGS: DYNAMICS AND NUMBER THEORY HEE OH Abstract. We give an overview of various counting problems for Apol- lonian circle packings, which turn out to be related to problems in dy- namics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi lec- tures given at RIMS, Kyoto in the fall of 2013. Contents 1. Counting problems for Apollonian circle packings 1 2. Hidden symmetries and Orbital counting problem 6 3. Counting, Mixing, and the Bowen-Margulis-Sullivan measure 9 4. Integral Apollonian circle packings 14 5. Expanders and Sieve 19 References 24 1. Counting problems for Apollonian circle packings An Apollonian circle packing is one of the most of beautiful circle packings whose construction can be described in a very simple manner based on an old theorem of Apollonius of Perga: Theorem 1.1 (Apollonius of Perga, 262-190 BC). Given 3 mutually tangent circles in the plane, there exist exactly two circles tangent to all three. Figure 1. Pictorial proof of the Apollonius theorem 1
Transcript of APOLLONIAN CIRCLE PACKINGS: DYNAMICS AND …apollonian circle packings: dynamics and number ... 121...
APOLLONIAN CIRCLE PACKINGS: DYNAMICS AND
NUMBER THEORY
HEE OH
Abstract. We give an overview of various counting problems for Apol-lonian circle packings, which turn out to be related to problems in dy-namics and number theory for thin groups. This survey article is anexpanded version of my lecture notes prepared for the 13th Takagi lec-tures given at RIMS, Kyoto in the fall of 2013.
Contents
1. Counting problems for Apollonian circle packings 12. Hidden symmetries and Orbital counting problem 63. Counting, Mixing, and the Bowen-Margulis-Sullivan measure 94. Integral Apollonian circle packings 145. Expanders and Sieve 19References 24
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 anold theorem 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.
Figure 1. Pictorial proof of the Apollonius theorem
1
2 HEE OH
Figure 2. Possible configurations of four mutually tangent circles
Proof. We give a modern proof, using the linear fractional transformationsof PSL2(C) on the extended complex plane C = C∪∞, known as Mobiustransformations: (
a bc d
)(z) =
az + b
cz + d,
where a, b, c, d ∈ C with ad− bc = 1 and z ∈ C ∪ ∞. As is well known, a
Mobius transformation maps circles in C to circles in C, preserving anglesbetween them. (In the whole article, a line in C is treated as a circle in C).In particular, 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 ∈ PSL2(C) be an elementwhich maps p to ∞. Then g maps C1 and C2 to two circles tangent at ∞,that is, two parallel lines, and g(C3) is a circle tangent to these parallel lines.In the configuration of g(C1), g(C2), g(C3) (see Fig. 1), it is clear that thereare precisely two circles, say, D and D′ tangent to all three g(Ci), 1 ≤ i ≤ 3.Using g−1, which is again a Mobius transformation, it follows that g−1(D)and g−1(D′) are precisely those two circles tangent to C1,C2,C3.
In order to construct an Apollonian circle packing, we begin with fourmutually tangent circles in the plane (see Figure 2 for possible configura-tions) and keep adding newer circles tangent to three of the previous circlesprovided by Theorem 1.1. Continuing this process indefinitely, we arrive atan infinite circle packing, called an Apollonian circle packing.
APOLLONIAN PACKINGS 3
2
23
2
2315
6
6
32
2315
38
38
35
623 14
11
623 14
11
3 15
6
6 2
2315
3871114
167
374
371
222455
554659
219447
546
654
110287762
546
867
218647
362
683
323927
858
642
107279738
846
531
210623
662347318911
842 635
3871222455
554659
114167
374
371
219447 654
546
110287546
762
867
218647
362
683
323927642
858
107279846
738
531
318911
842 635
210623
662347
3563198
407
590
491
198407
590
491
99143326
326
102263498
798
699
302863
602
803
203599
338
638
102263798
498
699
302863
602
803
203599
638
338
623
5087
134
278
275
150383
302443
147375
294
438
62 119362
194
371
170447330
507
179471522
354
59111
338 350
179
162423
486
315
174455
506 347 1447122231
330
36398167
290
299
131255
378
354
2687182
218
251
42143
62
131
75231
150
186
35111314290
22790279
234
171
66215
194
107
1139
98183
294
267
110215314
299
83143
242254
1863134
182155
54167
110
131
2795
3886
3095266
194251
74231
138
195
59191
170
98
623 62 119
194362
371
170447330
507
179471354
522
5087 278
134
275
150383
302443
147375 438
294
59111
350338
179
174455
506 347162
423
486
315144798167
290
299
122231
330
363
131255
354
378
2687218
182251
42143
62
131
75231
186
150
35111290
314
227
66215
194
107
90279
234
171
1139
110215314
299
98183
294
267
83143
254242
3095
194
266
251
74231
138
195
59191 98
170
1863
182
134
155
54167
110
131
2795
86
38
3 1538
71114
222
219
110287
218
323
107279
210318
3871222
114
219
110287
218
323
107279
318210
35 63198
198
99
102263302
203
102263302
203
623
5087
150
14762119
170
179
59111
162174
14 47122
98
131
26 87
42
75
3511190
66
1139
98110
83
1863
54
27
3095
74
59
62362119
170
179
5087
150
147
59111
174162
14 47 98122
131
26 87
42
75
35111
66
90
1139
11098
83
3095
74
59
1863
54
27
Figure 3. First few generations
1 141224406084
201201
129316
268385129
316
268385
73148
432
249465
184492345
561
217580616
43373148432
465
249
184492 561
345
217580
616
433
3388232444609
705
169276
504537
265532 760
697
64192388
481577
105336
156345
201592412505
97280808745
553
256768
681 489
193600
5763213388232444 705
609
265532760
697
169276
504 537
64192388
577481
201592 412505
105336
156345
97280808
553
745
256768
489 681
193600
576 321
9 1652108
184313
337
129348
240393
153412
304
433
2536124
49121
84204177
249
81196244
16949100
316
289169
148396 417
297
121324
372
225
2872136220409
433
193372
504585
217436 568
625
57168424
337505
96304145
313
177520448
361
81232616
673 457
160496481
265
217648576
417
2564196396
561 513
169324
516
441 121196388
361
76216424625
577201600384
537
153472
256
457
49144436
289361
156456
321
393
81256268
121
91652108
184
337
313
153412
304
433
129348
240 393
49100316
169289
148396297
417
121324372
225
2536124121
49
84204249
177
81196
244
169
2872136220 433
409
217436568
625
193372
504 585
81232616
457 673
160496
265
481
217648 576
417
57168424
505337
96304313
145
177520 448
361
2564
196396
513 561
121196388
361 169324516
441
76216
424
577
625
153472
256
457
201600 384
537
49144
436
361289
156456
393
321
81256
268
121
41224406084
201201
129268
316
385
129268
316
385
73184492345
561
148432
249465
217616
58043373184492 561
345
148432
465
249
217616
580 433
3364192388
481577
105156
336
345
201412 592
505
88232444609
705
169504276
537
265760532
697
97256768
681 489
280808745
553
193576600
3213364192388
577481
201412592
505105156
336
345
88232444 705
609
265760 532
697
169504
276
537
97256768
489 681
280808
553
745
193576
600
321
92872136
220
409
433
193504372
585
217568436
625
5796304
145
313
168424
337505
177448 520
361
81160496481
265
232616
673 457
217576 648
417
1652
108184313
337 129240
348
393
153304
412433
2584204
177
249
36124
4912181244
19616949148396 417
297
100316
289169121372
324 225
2576
216424625
577201384 600
537
153256
472457
64196396
561 513
169516
324 441
121388
19636149156456
321
393
144436
28936181268
256
121
92872136220 433
409
217568 436
625
193504
372
585
81160496
265
481
232616
457 673
217576648
417
5796304313
145
168424
505337
177448520
361
1652108184337
313
153304
412433
129240
348
393
49148396297
417
100316
169289
121372
324225
2584204249
177
36124
12149 81244
196 169 2576
216424
577
625
153256
472 457
201384600
537
64196396
513 561
121388196 361 169
516
324441 49156456
393
321
144436
361289 81268
256
121
1412244060
129 129
73148
184
217 73148
184
217
3388232
169
265
64192
105201
97280
256193 33
88232265
169
64192
201
105
97280
256193
91652108129
153
253684
81 49100148
121
2872136
193
217
57168
96
177
81232
160
217
2564
196
169121
76216
201
153
49144
156
81
9 1652108
153129
49100
148
121
253684
81
2872136217
193
81232160
217
57168
96
177
2564196
121169
76216
153
201
49144
156
81
412244060
129 129
73184
148217 73
184
148217
3364192
105201
88232
169
265
97256
280
193 3364192201
105
88232265
169
97256
280
193
92872136
193
217
5796
168177
81160
232
217
1652108
129
153
2584
36 81 49148
100121 2576
216 201
153
64196
169121 49
156
144 81
92872136217
193
81160
232
217
5796
168177
1652
108
153
129
49148
100 121 2584
36812576
216
153
201
64196
12116949
156
14481
14122440
7373
3388
649733
88
6497
9 1652
2549
2872
57
81
2564
76
49916
52
4925
287281
57
2564
76
49
4122440
7373
336488
973364
88
97
92872
57
81
1652
25492576
6449
92872
81
57
1652
4925 2576
64 49
141224
33 33
916
28
25 9 16
28
25
41224
33 33
928
16 25 928
1625
1412
99
412
99
14
4 11412244060
129 129
73148
184
217 73148
184
217
3388232
169
265
64192
105201
97280
256193 33
88232265
169
64192
201
105
97280
256193
91652108129
153
253684
81 49100148
121
2872136
193
217
57168
96
177
81232
160
217
2564
196
169121
76216
201
153
49144
156
81
9 1652108
153129
49100
148
121
253684
81
2872136217
193
81232160
217
57168
96
177
2564196
121169
76216
153
201
49144
156
81
412244060
129 129
73184
148217 73
184
148217
3364192
105201
88232
169
265
97256
280
193 3364192201
105
88232265
169
97256
280
193
92872136
193
217
5796
168177
81160
232
217
1652108
129
153
2584
36 81 49148
100121 2576
216 201
153
64196
169121 49
156
144 81
92872136217
193
81160
232
217
5796
168177
1652
108
153
129
49148
100 121 2584
36812576
216
153
201
64196
12116949
156
14481
14122440
7373
3388
649733
88
6497
9 1652
2549
2872
57
81
2564
76
49916
52
4925
287281
57
2564
76
49
4122440
7373
336488
973364
88
97
92872
57
81
1652
25492576
6449
92872
81
57
1652
4925 2576
64 49
141224
33 33
916
28
25 9 16
28
25
41224
33 33
928
16 25 928
1625
1412
99
412
99
14
41Figure 4. Unbounded Apollonian circle packing
Figure 3 shows the first few generations of this process, where each cir-cle is labeled with its curvature (= the reciprocal of its radius) with thenormalization that the greatest circle has radius one.
If we had started with a configuration containing two parallel lines, wewould have arrived at an unbounded Apollonian circle packing as in Figure4. There are also other unbounded Apollonian packings containing eitheronly one line or no line at all; but it will be hard to draw them in a paper withfinite size, as circles will get enormously large only after a few generations.
For a bounded Apollonian packing P, there are only finitely many circlesof radius bigger than a given number. Hence the following counting functionis well-defined for any T > 0:
NP(T ) := #C ∈ P : curv(C) ≤ T.
Question 1.2. • Is there an asymptotic formula of NP(T ) as T →∞?• If so, can we compute?
The study of this question involves notions related to metric propertiesof the underlying fractal set called a residual set:
Res(P) := ∪C∈PC,
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 α. The notion of the Hausdorff dimen-sion was first given by Hausdorff in 1918. To explain its definition, we firstrecall the notion of the Hausdorff measure (cf. [36]):
4 HEE OH
Definition 1.3. Let s ≥ 0 and F be any subset of Rn. The s-dimensionalHausdorff measure of F is defined by
Hs(F ) := limε→0
(inf
∑d(Bi)
s : F ⊂ ∪iBi, d(Bi) < ε)
where d(Bi) is the diameter of Bi.
For s = n, it is the usual Lebesgue measure of Rn, up to a constantmultiple. It can be shown that as s increases, the s-dimensional Hausdorffmeasure of F will be ∞ up to a certain value and then jumps down to 0.The Hausdorff dimension of F is this critical value of s:
dimH(F ) = sups : Hs(F ) =∞ = infs : Hs(F ) = 0.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 ≤ α ≤ 2.• α is independent of P: any two Apollonian packings are equivalent
to each other by a Mobius transformation which maps three tangentpoints of one packing to three tangent points of the other packing.• The precise value of α is unknown, but approximately, α = 1.30568(8)
due to McMullen [37].
In particular, Res(P) is much bigger than a countable union of circles (asα > 1), but not too big in the sense that its Lebesgue area is zero (as α < 2).
The first counting result for Apollonian packings is due to Boyd in 1982[7]:
Theorem 1.4 (Boyd).
limT→∞
logNP(T )
log T= α.
Boyd asked in [7] whether NP(T ) ∼ c · Tα as T →∞, and wrote that hisnumerical experiments suggest this may be false and perhaps
NP(T ) ∼ c · Tα(log T )β
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 · Tα as T →∞.
Theorem 1.6 (Lee-O. [32]). There exists η > 0 such that for any boundedApollonian packing P,
NP(T ) = cP · Tα +O(Tα−η).
APOLLONIAN PACKINGS 5
Vinogradov [58] has also independently obtained Theorem 1.6 with aweaker error term.
For an unbounded Apollonian packing P, we have NP(T ) =∞ in general;however we can modify our counting question so that we count only thosecircles contained in a fixed curvilinear triangle R whose sides are given bythree mutually tangent circles.
SettingNR(T ) := #C ∈ R : curv(C) ≤ T <∞,
we have shown:
Theorem 1.7 (O.-Shah [45]). For a curvilinear triangle R of any Apollo-nian packing P, there exists a constant cR > 0 such that
NR(T ) ∼ cR · Tα as T →∞.
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 →∞. To formulate thisquestion precisely, for any bounded region E ⊂ C, we set
NP(T,E) := #C ∈ P : C ∩ E 6= ∅, curv(C) ≤ T.Then the question on the asymptotic distribution of circles in P amounts
to searching for a locally finite Borel measure ωP on the plane C satisfyingthat
limT→∞
NT (P, E)
Tα= ωP(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 [45]). For any bounded Borel E ⊂ C with smoothboundary,
NP(T,E) ∼ cA · HαP(E) · Tα as T →∞where HαP denotes the α-dimensional Hausdorff measure of the set Res(P)and 0 < cA <∞ is a constant independent of P.
In general, dimH(F ) = s does not mean that the s-dimensional Hausdorffmeasure Hs(F ) is non-trivial (it could be 0 or ∞). But on the residualset Res(P) of an Apollonian packing, HαP is known to be locally finite and
6 HEE OH
Figure 5. Dual circles
its support is precisely Res(P) by Sullivan [57]; hence HαP(E) < ∞ for Ebounded and 0 < HαP(E) if E ∩ Res(P) 6= ∅.
Though the Hausdorff dimension and the packing dimension are equalto each other for Res(P), the packing measure is locally infinite ([57], [39])which indicates 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 α-dimensional Hausdorff measure on Res(P):for any bounded Borel subsets E1, E2 ⊂ C with smooth boundaries and withE2 ∩ Res(P) 6= ∅,
limT→∞
NP(T,E1)
NP(T,E2)=HαP(E1)
HαP(E2).
Apollonian constant. Observe that the constant cA in Theorem 1.8 isgiven by
cA = limT→∞
NP(T,E)
Tα · HαP(E)
for any Apollonian circle packing P and any E with E ∩ Res(P) 6= ∅. Inparticular,
cA = limT→∞
NP(T )
Tα · HαP(Res(P))
for any bounded Apollonian circle packing P.
Definition 1.9. We propose to call 0 < cA <∞ the Apollonian constant.
Problem 1.10. What is cA? (even approximation?)
Whereas all other terms in the asymptotic formula of Theorem 1.7 canbe described using the metric notions of Euclidean plane, our exact formulaof cA involves certain singular measures of an infinite-volume hyperbolic 3manifold, indicating the intricacy of the precise counting problem.
APOLLONIAN PACKINGS 7
2. Hidden symmetries and Orbital counting problem
Hidden symmetries. The key to our approach of counting circles in anApollonian packing lies in the fact that
An Apollonian circle packing has lots of hidden symmetries.
Explaining these hidden symmetries will lead us to explain the relevance ofthe packing with a Kleinian group, called the (geometric) Apollonian group.
Fix 4 mutually tangent circles C1, C2, C3, C4 in P and consider their dualcircles C1, · · · , C4, that is, Ci is the unique circle passing through the threetangent points among Cj ’s for j 6= i. In Figure 5, the solid circles representCi’s and the dotted circles are their dual circles. Observe that invertingwith respect to a dual circle preserves the three circles that it meets perpen-dicularly and interchanges the two circles which are tangent to those threecircles.
Definition 2.1. The inversion with respect to a circle of radius r centered at
a maps x to a+ r2
|x−a|2 (x− a). The group Mob(C) of Mobis transformations
in C is generated by inversions with respect to all circles in C.
The geometric Apollonian group A := AP associated to P is generatedby the four inversions with respect to the dual circles:
A = 〈τ1, τ2, τ3, τ4〉 < Mob(C)
where τi denotes the inversion with respect to Ci. Note that PSL2(C) is a
subgroup of Mob(C) of index two; we will write Mob(C) = PSL2(C)±. TheApollonian groupA is a Kleinian group (= a discrete subgroup of PSL2(C)±)and satisfies
• P = ∪4i=1A(Ci), that is, inverting the initial four circles in P with
respect to their dual circles generates the whole packing P;• Res(P) = Λ(A) where Λ(A) denotes the limit set of A, which is the
set of all accumulation points of an orbit A(z) for z ∈ C.
In order to explain how the hyperbolic geometry comes into the picture,it is most convenient to use the upper-half space model for hyperbolic 3
8 HEE OH
Figure 6. Whitehead Link
space H3: H3 = (x1, x2, y) : y > 0. The hyperbolic metric is given
by ds =
√dx21+dx22+dy2
y and the geometric boundary ∂∞(H3) is naturally
identified with C. Totally geodesic subspaces in H3 are vertical lines, verticalcircles, vertical planes, and vertical hemispheres.
The Poincare extension theorem gives an identification Mob(C) with the
isometry group Isom(H3). Since Mob(C) is generated by inversions with
respect to circles in C, the Poincare extension theorem is determined bythe correspondence which assigns to an inversion with respect to a circle Cin C the inversion with respect to the vertical hemisphere in H3 above C.An inversion with respect to a vertical hemisphere preserves the upper halfspace, as well as the hyperbolic metric, and hence gives rise to an isometryof H3.
The Apollonian group A = 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\H3 is an infinitevolume hyperbolic 3-manifold and has a fundamental domain with finitelymany sides; such a manifold is called a geometrically finite manifold.
Connection with the Whitehead link. The Apollonian manifold A\H3
can also be constructed from the Whitehead link complement. To explainthe connection, consider the group, say, A∗ generated by 8 inversions withrespect to four mutually tangent circles as well as their four dual circles.Then the group A∗ has a regular ideal hyperbolic octahedron as a funda-mental domain in H3, and is commensurable to the Picard group PSL2(Z[i]),up to a conjugation, which is a lattice in PSL2(C). The quotient orbifoldA∗\H3 is commensurable to the Whitehead link complement S3 −W (seeFigure 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 Wand pierced twice by the other component), which is totally geodesic andwhose fundamental group is conjugate to the congruence subgroup Γ(2) of
APOLLONIAN PACKINGS 9
PSL2(Z) of level 2. If we cut the manifold S3 −W open along this totallygeodesic surface Γ(2)\H2, we get a finite volume hyperbolic manifold withtotally geodesic boundary, whose fundamental group is the Apollonian groupA. We thank Curt McMullen for bringing this beautiful relation with theWhitehead link to our attention.
Orbital counting problem in PSL2(R)\PSL2(C). Observe that the num-ber of circles in an Apollonian packing P of curvature at most T is sameas the number of the vertical hemispheres above circles in P of Euclideanheight 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 in P whichintersects the cylindrical region
ET := (x1, x2, y) ∈ H3 : x1 + ix2 ∈ E, T−1 ≤ y ≤ r0 (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 (in-cluding planes), the space of vertical hemispheres in H3 can be identifiedwith the homogeneous space PSL2(R)\PSL2(C). Since P consists of finitelymany A-orbits of circles in C, which corresponds to finitely many A-orbitsof points in PSL2(R)\PSL2(C), understanding the asymptotic formula ofNP(T,E) is a special case of the following more general counting problem:letting G = PSL2(C) and H = PSL2(R), for a given sequence of growingcompact subsets BT in H\G and a discrete A-orbit v0A in H\G,
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-understood 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 thelattice point counting problem in Euclidean space. Let G = R3, Γ = Z3 andlet BT := x ∈ R3 : ‖x‖ ≤ T be the Euclidean ball of radius T centered atthe origin. In showing the well-known fact
#Z3 ∩BT ∼4π
3T 3,
one way is to count the Γ-translates of a fundamental domain, say F :=[−1
2 ,12) × [−1
2 ,12) × [−1
2 ,12) contained in BT , since each translate γ + F
contains precisely one point, that is, γ, from Γ. We have
Vol(BT−1)
Vol(F)≤ #γ + F ⊂ BT−1 : γ ∈ Z3
≤ #Z3 ∩BT ≤ #γ + F ⊂ BT+1 : γ ∈ Z3 ≤ Vol(BT+1)
Vol(F), (3.1)
10 HEE OH
SinceVol(BT±1)
Vol(F) = 4π3 (T ± 1)3, we obtain that
#Z3 ∩BT =4π
3T 3 +O(T 2).
This easily generalizes to the following: for any discrete subgroup Γ in R3
and a sequence BT of compact subsets in R3, we have
#Γ ∩BT =Vol(BT )
Vol(Γ\R3)+O(Vol(BT )1−η)
provided
• Vol(Γ\R3) <∞;• Vol(unit neighborhood of ∂(BT )) = O(Vol(BT )1−η) for some η > 0.
We have used here that the volume in R3 is computed with respect to theLebesgue measure which is clearly left Γ-invariant so that it makes sense to
write Vol(Γ\R3), and that the ratio Vol(unit neighborhood of ∂(BT ))Vol(BT ) tends to 0 as
T →∞.
Hyperbolic lattice point counting We now consider the hyperbolic lat-tice counting problem for H3. Let G = PSL2(C) and Γ be a torsion-free,co-compact, discrete subgroup of G. The group G possess a Haar measureµG which is both left and right invariant under G, in particular, it is left-invariant under Γ. By abuse of notation, we use the same notation µG forthe induced measure on Γ\G. Fix o = (0, 0, 1) so that g 7→ g(o) inducesan isomorphism of H3 with G/PSU(2) and hence µG also induces a leftG-invariant measure on H3, which will again be denoted by µG. Considerthe hyperbolic ball BT = x ∈ H3 : d(o, x) ≤ T where d is the hyperbolicdistance in H3. Then, for a fixed fundamental domain F for Γ in H3 whichcontains o in its interior, we have inequalities similar to (3.1):
Vol(BT−d)
Vol(Γ\G)≤ #γ(F) ⊂ BT−d : γ ∈ Γ
≤ #Γ(o) ∩BT ≤ #γ(F) ⊂ BT+d : γ ∈ Γ ≤ Vol(BT+d)
Vol(Γ\G)(3.2)
where d is the diameter of F and the volumes Vol(BT±d) and Vol(Γ\G) arecomputed with respect to µG on H3 and Γ\G respectively.
If we had Vol(BT−d) ∼ Vol(BT+d) as T → ∞ as in the Euclidean case,
we would be able to conclude from here that #Γ(o) ∩ BT ∼ Vol(BT )Vol(Γ\G) 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+d) is not true.This suggests that the above inequality (3.2) gives too crude estimation ofthe edge effect arising from the intersections of γ(F)’s with BT near theboundary of BT . It turns out that the mixing phenomenon of the geodesicflow on the unit tangent bundle T1(Γ\H3) with respect to µG precisely clearsout the fuzziness of the edge effect. The mixing of the geodesic flow follows
APOLLONIAN PACKINGS 11
from the following mixing of the frame flow, or equivalently, the decay of
matrix coefficients due to Howe and Moore [28]: Let at :=
(et/2 0
0 e−t/2
).
Theorem 3.3 (Howe-Moore). Let Γ < G be a lattice. For any ψ1, ψ2 ∈Cc(Γ\G),
limt→∞
∫Γ\G
ψ1(gat)ψ2(g)dµG(g) =1
µG(Γ\G)
∫Γ\G
ψ1dµG ·∫
Γ\Gψ2dµG.
Indeed, using this mixing property of the Haar measure, there are nowvery well established counting result due to Duke-Rudnick-Sarnak [15], andEskin-McMullen[17]): we note that any symmetric subgroup of G is locallyisomorphic to SL2(R) or SU(2).
Theorem 3.4 (Duke-Rudnick-Sarnak, Eskin-McMullen). Let H be a sym-metric subgroup of G and Γ < G a lattice such that µH(Γ∩H\H) <∞, i.e.,H ∩Γ is a lattice in H. Then for any well-rounded sequence BT of compactsubsets in H\G and a discrete Γ-orbit [e]Γ, we have
#[e]Γ ∩BT ∼µH((H ∩ Γ)\H)
µG(Γ\G)·Vol(BT ) as T →∞.
Here the volume of BT is computed with respect to the invariant measureµH\G on H\G which satisfies µG = µH ⊗ µH\G locally.
A sequence BT ⊂ H\G is called well-rounded with respect to a measureµ on H\G if the boundaries of BT are µ-negligible, more precisely, if for allsmall ε > 0, the µ-measure of the ε-neighborhood of the boundary of BT isO(ε · µ(BT )) as T →∞.
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 Γ < G = PSL2(C) is not a lattice, thatis, µG(Γ\G) = ∞. It turns out that as long as we have a left Γ-invariantmeasure, say, µ on G, satisfying
• µ(Γ\G) <∞;• µ is the mixing measure for the frame flow on Γ\G,
then the above heuristics of comparing the counting function for #Γ(o)∩BTto the volume µ(BT ) can be made into a proof.
For what kind of discrete groups Γ, do we have a left-Γ-nvariant measureon G satisfying these two conditions? Indeed when Γ is geometrically finite,the Bowen-Margulis-Sullivan measure mBMS on Γ\G satisfies these proper-ties. Moreover when Γ is convex cocompact (that is, geometrically finitewith no parabolic elements), the Bowen-Margulis-Sullivan measure is sup-ported on a compact subset of Γ\G. Therefore Γ acts co-compactly in theconvex hull CH(Λ(Γ)) of the limit set Λ(Γ); recall that Λ(Γ) is the set of allaccumulation points of Γ-orbits on the boundary ∂(H3). Hence if we denoteby F0 a compact fundamental domain for Γ in CH(Λ(Γ)), the inequality
12 HEE OH
(3.2) continues to hold if we replace the fundamental domain F of Γ in H3
by F0 and compute the volumes with respect to mBMS:
mBMS(BT−d0)
mBMS(Γ\G)≤ #γ(F0) ⊂ BT−d0 : γ ∈ Γ
≤ #Γ(o) ∩BT ≤ #γ(F0) ⊂ BT+d0 : γ ∈ Γ ≤ mBMS(BT+d0)
mBMS(Γ\G)(3.5)
where mBMS is the projection to H3 of the lift of mBMS to G and d0 is thediameter of F0. This suggests a heuristic expectation:
#Γ(o) ∩BT ∼mBMS(BT )
mBMS(Γ\G)
which turns out to be true.We denote by δ the Hausdorff dimension of Λ(Γ) which is known to be
equal to the critical exponent of Γ. Patterson [46] and Sullivan[56] con-structed a unique geometric probability measure νo on ∂(H3) satisfying thatfor any γ ∈ Γ, γ∗νo is absolutely continuous with respect to νo and for anyBorel subset E,
νo(γ(E)) =
∫E
(d(γ∗νo)
dνo
)δdνo.
This measure νo is called the Patterson-Sullivan measure viewed from o ∈H3. Then the Bowen-Margulis-Sullivan measure mBMS on T1(H3) is givenby
dmBMS(v) = f(v) dνo(v+)dνo(v
−)dt
where v± ∈ ∂(H3) are the forward and the backward endpoints of the geo-desic determined by v and t = βv−(o, v) measures the signed distance of thehoropsheres based at v− passing through o and v. The density function f isgiven by f(v) = eδ(βv+ (o,v)+βv− (o,v)) so that mBMS is left Γ-invariant Clearly,the support of mBMS is given by the set of v with v± ⊂ Λ(Γ). Noting thatT1(H3) is isomorphic to G/M where M = diag(eiθ, e−iθ), we will extendmBMS to an M -invariant measure on G. We use the same notation mBMS
to denote the measure induced on Γ\G.
Theorem 3.6. For Γ geometrically finite and Zariski dense,
(1) Finiteness: mBMS(Γ\G) <∞(2) Mixing: For any ψ1, ψ2 ∈ Cc(Γ\G), as t→∞,∫
Γ\Gψ1(gat)ψ2(g)dmBMS(g)→ 1
mBMS(Γ\G)
∫Γ\G
ψ1dmBMS
∫Γ\G
ψ2dmBMS.
The finiteness result (1) is due to Sullivan [56] and the mixing result (2)for frame flow is due to Flaminio-Spatzier [21] and Winter [59] based on thework of Rudolph [50] and Babillot [8].
In order to state an analogue of Theorem 3.4 for a general geometricallyfinite group, we need to impose a condition on (H ∩ Γ)\H analogous to
APOLLONIAN PACKINGS 13
the finiteness of the volume µH(Γ ∩H\H). In [44], we define the so calledskinning measure µPS
H on (Γ ∩H)\H, which is intuitively the slice measureon H of mBMS. We note that µPS
H depends on Γ, not only on H ∩ Γ. Afiniteness criterion for µPS
H is given in [44]. The following is obtained in [44]non-effectively and [38] effectively.
Theorem 3.7 (O.-Shah, Mohammadi-O.). Let H be a symmetric subgroupof G and Γ < G a geometrically finite and Zariski dense subgroup. Supposethat the skinning measure of H ∩ Γ\H is finite, i.e., µPS
H (Γ ∩ H\H) < ∞.Then there exists an explicit locally finite Borel measure MH\G on H\Gsuch that for any well-rounded sequence BT of compact subsets in H\G withrespect to MH\G and a discrete Γ-orbit [e]Γ, we have
#[e]Γ ∩BT ∼µPSH ((H ∩ Γ)\H)
mBMS(Γ\G)· MH\G(BT ) as T →∞.
A special case of this theorem implies Theorem 1.8, modulo the compu-tation of the measure MH\G(ET ) where ET is given in (2.2). We mentionthat in the case when the critical exponent δ of Γ is strictly bigger than 1,both Theorem 3.7 and Theorem 1.8 can be effectivized by [38].
The reason that we have the α-dimensional Hausdorff measure in thestatement of Theorem 1.8 is because the slice measure of mBMS on eachhorizontal plane is the Patterson-Sullivan measure multiplied with a correctdensity function needed for the Γ-invariance, which turns out to coincidewith the δ-dimensional Hausdorff measure on the limit set of Λ(Γ) when allcusps of Γ are of rank at most 1, which is the case for the Apollonian group.
Counting problems for Γ-orbits in H\G are technically much more in-volved when H is non-compact than when H is compact, and relies onunderstanding the asymptotic distribution of Γ\ΓHat in Γ\G as t → ∞.When H = PSL2(R), the translate Γ\ΓHat corresponds to the orthogonaltranslate of a totally geodesic surface for time t, and we showed that, afterthe correct scaling of e(2−δ)t, Γ\ΓHat becomes equidistributed in Γ\G withrespect to the Burger-Roblin measure mBR, which is the unique non-trivialergodic horospherical invariant measure on Γ\G. We refer to [42], [43], [45]for more details.
More circle packings. This viewpoint of approaching Apollonian circlepackings via the study of Kleinian groups allows us to deal with more gen-eral circle packings, provided they are invariant under a non-elementarygeometrically 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-empty totally geodesic boundary. Then
• Γ := π1(X) is a geometrically finite Kleinian group;• By developing X in the upper half space H3, the domain of discon-
tinuity Ω(Γ) := C−Λ(Γ) consists of the disjoint union of open disks
14 HEE OH
Figure 7. Sierpinski curve
(corresponding to the boundary components of the universal cover
X).
Set P to be the union of circles which are boundaries of the disks in Ω(Γ).In this case, Res(P) defined as the closure of all circles in P is equal to thelimit set Λ(Γ).
In section 2, we explained how Apollonian circle packings can be describedin this way.
Figure 7, due to McMullen, is also an example of a circle packing ob-tained in this way, here the symmetry group Γ is the fundamental groupof a compact hyperbolic 3-manifold with totally geodesic boundary beinga compact surface of genus two. This limit set is called a Sierpinski curve,being homeomorphic to the well-known Sierpinski Carpet.
Many more pictures of circle packings constructed in this way can befound in the book ”Indra’s pearls” by Mumford, Series and Wright (Cam-bridge Univ. Press 2002).
For P constructed in Example 3.8, we define as before NP(T,E) :=#C ∈ P : C ∩ E 6= ∅, curv(C) ≤ T for any bounded Borel subset Ein C.
Theorem 3.9 (O.-Shah, [44]). There exist a constant cΓ > 0 and a locallyfinite Borel measure ωP on Res(P) such that for any bounded Borel subsetE ⊂ C with ωP(∂(E)) = 0,
NP(T,E) ∼ cΓ · ωP(E) · T δ as T →∞
APOLLONIAN PACKINGS 15
where δ = dimH(Res(P)). Moreover, if Γ is convex cocompact or if thecusps of Γ have rank at most 1, then ωP coincides with the δ-dimensionalHausdorff measure on Res(P).
We refer to [44] 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 hasintegral curvature. Does there exist any integral P? The answer is positivethanks to the following beautiful theorem of Descartes:
Theorem 4.1 (Descartes 1643, [13]). A quadruple (a, b, c, d) is the curva-tures of four mutually tangent circles if and only if it satisfies the quadraticequation:
2(a2 + b2 + c2 + d2) = (a+ b+ c+ d)2.
In the above theorem, we ask circles to be oriented so that their interi-ors are disjoint with each other. For instance, according to this rule, thequadruple of curvatures of four largest four circles in Figure 3 is (−1, 2, 2, 3)or (1,−2,−2,−3), for which we can easily check the validity of the Descartestheorem: 2((−1)2 + 22 + 22 + 32) = 36 = (−1 + 2 + 2 + 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 circleswill then have positive curvatures.
Given three mutually tangent circles of curvatures a, b, c, the curvatures,say, d and d′, of the two circles tangent to all three must satisfy 2(a2 + b2 +c2 + d2) = (a+ b+ c+ d)2 and 2(a2 + b2 + c2 + (d′)2) = (a+ b+ c+ d′)2 bythe Descartes theorem. By subtracting the first equation from the second,
16 HEE OH
2
2315
3871114
167
374
371
222455
554659
219447
546
654
110287762
546
867
218647
362
683
323927
858
642
107279738
846
531
210623
662347318911
842635
3871222455
554659
114167
374
371
219447 654
546
110287546
762
867
218647
362
683
323927642
858
107279846
738
531
318911
842635
210623
662347
3563198
407
590
491
198407
590
491
99143326
326
102263498
798
699
302863
602
803
203599
338
638
102263798
498
699
302863
602
803
203599
638
338
623
5087
134
278
275
150383
302443
147375
294
438
62 119362
194
371
170447330
507
179471522
354
59111
338350
179
162423
486
315
174455
506 347 1447122231
330
36398167
290
299
131255
378
354
2687182
218
251
42143
62
131
75231
150
186
35111314290
22790279
234
171
66215
194
107
1139
98183
294
267
110215314
299
83143
242254
1863134
182155
54167
110
131
2795
3886
3095266
194251
74231
138
195
59191
170
98
623 62 119
194362
371
170447330
507
179471354
522
5087 278
134
275
150383
302443
147375 438
294
59111
350338
179
174455
506 347162
423
486
315144798167
290
299
122231
330
363
131255
354
378
2687218
182251
42143
62
131
75231
186
150
35111290
314
227
66215
194
107
90279
234
171
1139
110215314
299
98183
294
267
83143
254242
3095
194
266
251
74231
138
195
59191 98
170
1863
182
134
155
54167
110
131
2795
86
38
3 1538
71114
222
219
110287
218
323
107279
210318
3871222
114
219
110287
218
323
107279
318210
35 63198
198
99
102263302
203
102263302
203
623
5087
150
14762119
170
179
59111
162174
14 47122
98
131
26 87
42
75
3511190
66
1139
98110
83
1863
54
27
3095
74
59
62362119
170
179
5087
150
147
59111
174162
14 47 98122
131
26 87
42
75
35111
66
90
1139
11098
83
3095
74
59
1863
54
27
1 141224406084
201201
129316
268385129
316
268385
73148
432
249465
184492
345
561
217580
616
43373148432
465
249
184492 561
345
217580
616
433
3388232
444
609
705
169276
504537
265532 760
697
64192388
481577
105336
156345
201592
412505
97280808745
553
256768
681 489
193600
5763213388232
444 705
609
265532760
697
169276
504 537
64192388
577481
201592
412505
105336
156345
97280
808
553
745
256768
489 681
193600
576 321
9 1652108
184313
337
129348
240393
153412
304
433
2536124
49121
84204177
249
81196244
16949100
316
289169
148396 417
297
121324
372
225
2872136220409
433
193372
504585
217436 568
625
57168424
337505
96304145
313
177520448
361
81232616
673 457
160496481
265
217648576
417
2564196396
561 513
169324
516
441 121196388
361
76216424
625
577201600384
537
153472
256
457
49144436
289361
156456
321
393
81256268
121
91652108
184
337
313
153412
304
433
129348
240393
49100316
169289
148396
297
417
121324372
225
2536124121
49
84204249
177
81196
244
169
2872136220 433
409
217436568
625
193372
504585
81232616
457 673
160496
265
481
217648 576
417
57168424
505337
96304313
145
177520448
361
2564
196396
513 561
121196388
361 169324516
441
76216
424
577
625
153472
256
457
201600 384
537
49144
436
361289
156456
393
321
81256
268
121
41224406084
201201
129268
316
385
129268
316
385
73184492345
561
148432
249465
217616
58043373184492 561
345
148432
465
249
217616
580 433
3364192388
481577
105156
336
345
201412592
505
88232444609
705
169504276
537
265760532
697
97256768
681 489
280808745
553
193576600
3213364192388
577481
201412592
505105156
336
345
88232444 705
609
265760 532
697
169504
276
537
97256768
489 681
280808
553
745
193576
600
321
92872136
220
409
433
193504372
585
217568436
625
5796304
145
313
168424
337505
177448
520
361
81160496481
265
232616
673 457
217576 648
417
1652
108184313
337129
240348
393
153304
412433
2584204
177
249
36124
4912181244
19616949148396 417
297
100316
289169121372
324 225
2576
216424625
577201384 600
537
153256
472457
64196396
561 513
169516
324 441
121388
19636149156456
321
393
144436
28936181268
256
121
92872136220 433
409
217568 436
625
193504
372
585
81160496
265
481
232616
457 673
217576648
417
5796304313
145
168424
505337
177448
520
361
1652108184337
313
153304
412433
129240
348
393
49148396297
417
100316
169289
121372
324225
2584204249
177
36124
12149 81244
196 169 2576
216424
577
625
153256
472 457
201384600
537
64196396
513 561
121388196 361 169
516
324441 49156456
393
321
144436
361289 81268
256
121
1412244060
129 129
73148
184
217 73148
184
217
3388232
169
265
64192
105201
97280
256193 33
88232265
169
64192
201
105
97280
256193
91652108129
153
253684
81 49100148
121
2872136
193
217
57168
96
177
81232
160
217
2564
196
169121
76216
201
153
49144
156
81
9 1652108
153129
49100
148
121
253684
81
2872136217
193
81232160
217
57168
96
177
2564196
121169
76216
153
201
49144
156
81
412244060
129 129
73184
148217 73
184
148217
3364192
105201
88232
169
265
97256
280
193 3364192201
105
88232265
169
97256
280
193
92872136
193
217
5796
168177
81160
232
217
1652108
129
153
2584
36 81 49148
100121 2576
216 201
153
64196
169121 49
156
144 81
92872136217
193
81160
232
217
5796
168177
1652
108
153
129
49148
100 121 2584
36812576
216
153
201
64196
12116949
156
14481
14122440
7373
3388
649733
88
6497
9 1652
2549
2872
57
81
2564
76
49916
52
4925
287281
57
2564
76
49
4122440
7373
336488
973364
88
97
92872
57
81
1652
25492576
6449
92872
81
57
1652
4925 2576
64 49
141224
33 33
916
28
25 9 16
28
25
41224
33 33
928
16 25 928
1625
1412
99
412
99
14
4 11412244060
129 129
73148
184
217 73148
184
217
3388232
169
265
64192
105201
97280
256193 33
88232265
169
64192
201
105
97280
256193
91652108129
153
253684
81 49100148
121
2872136
193
217
57168
96
177
81232
160
217
2564
196
169121
76216
201
153
49144
156
81
9 1652108
153129
49100
148
121
253684
81
2872136217
193
81232160
217
57168
96
177
2564196
121169
76216
153
201
49144
156
81
412244060
129 129
73184
148217 73
184
148217
3364192
105201
88232
169
265
97256
280
193 3364192201
105
88232265
169
97256
280
193
92872136
193
217
5796
168177
81160
232
217
1652108
129
153
2584
36 81 49148
100121 2576
216 201
153
64196
169121 49
156
144 81
92872136217
193
81160
232
217
5796
168177
1652
108
153
129
49148
100 121 2584
36812576
216
153
201
64196
12116949
156
14481
14122440
7373
3388
649733
88
6497
9 1652
2549
2872
57
81
2564
76
49916
52
4925
287281
57
2564
76
49
4122440
7373
336488
973364
88
97
92872
57
81
1652
25492576
6449
92872
81
57
1652
4925 2576
64 49
141224
33 33
916
28
25 9 16
28
25
41224
33 33
928
16 25 928
1625
1412
99
412
99
14
41Figure 8. Integral Apollonian packings
we obtain the linear equation:
d+ d′ = 2(a+ b+ c).
So, if a, b, c, d are integers, so is d′. Since the curvature of every circle fromthe second generation or later is d′ for some 4 mutually tangent circles ofcurvatures a, b, c, d from the previous generation, we deduce:
Theorem 4.2 (Soddy 1937). If the initial 4 circles in an Apollonian packingP have integral curvatures, P is integral.
Combined with Descartes’ theorem, for any integral solution of 2(a2+b2+c2+d2) = (a+b+c+d)2, there exists an integral Apollonian packing! Becausethe 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.
For a given integral Apollonian packing P, it is natural to inquire aboutits Diophantine properties such as
Question 4.3. • Are there infinitely many circles with prime curva-tures?• Which integers appear as curvatures?
We call P primitive, if g. c.dC∈P(curv(C)) = 1. We call a circle is primeif its curvature is a prime number, and a pair of tangent prime circles willbe called twin prime circles. There are no triplet primes of three mutuallytangent circles, all having odd prime curvatures.
Theorem 4.4 (Sarnak 07). There are infinitely many prime circles as wellas twin 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 infinitudeof prime numbers. In order to formulate what can be considered as ananalogue of the prime number theorem, we set
ΠT (P) := #prime C ∈ P : curv(C) ≤ T
APOLLONIAN PACKINGS 17
andΠ
(2)T (P) := #twin primes C1, C2 ∈ P : curv(Ci) ≤ T.
Using the sieve method based on heuristics on the randomness of Mobiusfunction, Fuchs and Sanden [21] conjectured:
Conjecture 4.5 (Fuchs-Sanden).
ΠT (P) ∼ c1NP(T )
log T; Π
(2)T (P) ∼ c2
NP(T )
(log T )2
where c1 > 0 and c2 > 0 can be given explicitly.
Based on the breakthrough of Bourgain, Gamburd, Sarnak [3] provingthat the Cayley graphs of congruence quotients of the integral Apolloniangroup form an expander family, together with Selberg’s upper bound sieve,we obtain upper bounds of true order of magnitude:
Theorem 4.6 (Kontorovich-O. [30]). For T 1,
• ΠT (P) Tα
log T ;
• Π(2)T (P) Tα
(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 saysthat the number of integers at most T arising as curvatures of circles inintegral P counted with multiplicity, is of order T 1.3.... So one may hopethat a positive density (=proportion) of integers arises as curvatures, asconjectured by Graham, Lagarias, Mallows, Wilkes, Yan (Positive densityconjecture) [24].
Theorem 4.7 (Bourgain-Fuchs [2]). For a primitive integral Apollonianpacking P,
#curv(C) ≤ T : C ∈ P T.
A stronger conjecture, called the Strong Density conjecture, of Grahamet al. says that every integer occurs as the value of a curvature of a circlein P, unless there are congruence obstructions. Fuchs [19] showed that theonly congruence obstructions are modulo 24, and hence the strong positivedensity conjecture (or the local-global principle conjecture) says that everysufficiently large integer which is congruent to a curvature of a circle in Pmodulo 24 must occur as the value of a curvature of some circle in P. Thisconjecture is still open, but there is now a stronger version of the positivedensity theorem:
Theorem 4.8 (Bourgain-Kontorovich [5]). For a primitive integral Apollo-nian packing P,
#curv(C) ≤ T : C ∈ P ∼ κ(P)
24· T
where κ(P) > 0 is the number of residue classes mod 24 of curvatures of P.
18 HEE OH
Improving Sarnak’s result on the infinitude of prime circles, Bourgainshowed that a positive fraction of prime numbers appear as curvatures inP.
Theorem 4.9 (Bourgain [6]).
#prime curv(C) ≤ T : C ∈ P T
log T.
Integral Apollonian group. In studying the Diophantine properties ofintegral Apollonian packings, we work with the integral Apollonian group,rather than the geometric Apollonian group which was defined in section 2.
We call a quadruple (a, b, c, d) a Descartes quadruple if it represents cur-vatures of four mutually tangent circles (oriented so that their interiors aredisjoint) in the plane. By Descartes’ theorem, any Descartes quadruple(a, b, c, d) lies on the cone Q(x) = 0, where Q denotes the so-called Descartesquadratic form
Q(a, b, c, d) = 2(a2 + b2 + c2 + d2)− (a+ b+ c+ d)2.
The quadratic form Q has signature (3, 1) and hence over the reals, theorthogonal group OQ is isomorphic to O(3, 1), which is the isometry groupof the hyperbolic 3-space H3.
We observe that if (a, b, c, d) and (a, b, c, d′) are Descartes quadruples,then d′ = −d+ 2(a+ b+ c) and hence (a, b, c, d′) = (a, b, c, d)S4 where
S1 =
−1 0 0 02 1 0 02 0 1 02 0 0 1
, S2 =
1 2 0 00 −1 0 00 2 1 00 2 0 1
,
S3 =
1 0 2 00 1 2 00 0 −1 00 0 2 1
, S4 =
1 0 0 20 1 0 20 0 1 20 0 0 −1
.
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 = 0. Moreover if wechoose a root quadruple vP from P, which consists of curvatures of fourlargest mutually tangent circles, any reduced word wn = vPSi1 · · ·Sin withSij ∈ S1, S2, S3, S4 is obtained from wn−1 = vPSi1 · · ·Sin−1 by changingprecisely one entry and this new entry is the maximum entry of wn, which isthe curvature of a precisely one new circle added at the n-th generation [24].This gives us the translation of the circle counting problem for a boundedApollonian packings as the orbital counting problem of an A-orbit in a coneQ = 0:
NT (P) = #v ∈ vPA : ‖v‖max ≤ T+ 3.
APOLLONIAN PACKINGS 19
The integral Apollonian group A is isomorphic to the geometric Apol-lonian group AP (the subgroup generated by four inversions with respectto the dual circles of four mutually tangent circles in P): there exists an
explicit isomorphism between the orthogonal group OQ and Mob(C) whichmaps the integral Apollonian group A to the geometric Apollonian groupAP . In particular, A is a subgroup OQ(Z) which is of infinite index andZariski dense in OQ. Such a subgroup is called a thin group. Diophantineproperties of an integral Apollonian packing is now reduced to the studyof Diophantine properties of an orbit of the thin group A. Unlike orbitsunder an arithmetic subgroup (subgroups of OQ(Z) of finite index) whichhas a rich theory of automorphic forms and ergodic theory, the study ofthin groups has begun very recently, but with a great success. In particular,the recent developments in expanders is one of key ingredients in studyingprimes or almost primes in thin orbits (see [6]).
5. Expanders and Sieve
All graphs will be assumed to be simple (no multiple edges and no loops)and connected in this section. For a finite k-regular graph X = X(V,E)with V = v1, · · · , vn the set of vertices and E the set of edges, the ad-jacency matrix A = (aij) is defined by aij = 1 if vi, vj ∈ E and aij = 0otherwise. Since A is a symmetric real matrix, it has n real eigenvalues:λ0(X) ≥ λ1(X) ≥ · · · ≥ λn−1(X). As X is simple and connected, thelargest eigenvalue λ0(X) is given by k and has multiplicity one.
Definition 5.1. A family of k-regular graphs Xi with ( #Xi → ∞) iscalled an expander family if there exists an ε0 > 0 such that
supiλ1(Xi) ≤ k − ε0.
Equivalently, Xi is an expander family if there exists a uniform positivelower bound for the Cheeger constant (or isoperimetric constant)
h(Xi) := min0<#W≤#Xi/2
#∂(W )
#W
where ∂(W ) means the set of edges with exactly one vertex in W . Notethat the bigger the Cheeger constant is, the harder it is to break the graphinto two pieces. Intuitively speaking, an expander family is a family ofsparse graphs (as the regularity k is fixed) with high connectivity properties(uniform lower bound for the Cheeger constants).
Although it was known that there has to be many expander families us-ing probabilistic arguments due to Pinsker, the first explicit constructionof an expander family is due to Margulis in 1973 [35] using the representa-tion theory of a simple algebraic group and automorphic form theory. Weexplain his construction below; strictly speaking, what we describe belowis not exactly same as his original construction but the idea of using the
20 HEE OH
representation theory of an ambient algebraic group is the main point of hisconstruction as well as in the examples below.
Let G be a connected simple non-compact real algebraic group definedover Q, with a fixed Q-embedding into SLN . Let G(Z) := G ∩ SLN (Z) andΓ < G(Z) be a finitely generated subgroup. For each positive integer q, theprincipal congruence subgroup Γ(q) of level q is defined to be γ ∈ Γ : γ = emod q.
Fix a finite symmetric generating subset S for Γ. Then S generates thegroup Γ(q)\Γ via the canonical projection. We denote byXq := C(Γ(q)\Γ, S)the Cayley graph of the group Γ(q)\Γ with respect to S, that is, vertices ofXq are elements of Γ(q)\Γ and two elements g1, g2 form an edge if g1 = g2sfor some s ∈ S. Then Xq is a connected k-regular graph for k = #S. Nowa key observation due to Margulis is that if Γ is of finite index in G(Z),or equivalently if Γ is a lattice in G, then the following two properties areequivalent: for any I ⊂ N,
(1) The family Xq : q ∈ I is an expander;(2) The trivial representation 1G is isolated in the sum ⊕q∈IL2(Γ(q)\G)
in the Fell topology of the set of unitary representations of G.
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 generatingsubset Q of G, there exists ε > 0 (independent of q ∈ I) such that any unitvector f ∈ L2(Γ(q)\G) satisfying maxq∈Q ‖q.f−f‖ < ε is G-invariant, i.e., aconstant. Briefly speaking, it follows almost immediately from the definitionof an expander family that the family Xq is an expander if and only ifthe trivial representation 1Γ of Γ is isolated in the sum ⊕qL2(Γ(q)\Γ). Onthe other hand,the induced representation of 1Γ from Γ to G is L2(Γ\G),which contains the trivial representation 1G, if Γ is a lattice in G. Therefore,by the continuity of the induction process, the weak containment of 1Γ in⊕qL2(Γ(q)\Γ) implies the weak-containment of 1G in ⊕qL2(Γ(q)\G), whichexplains why (2) implies (1).
The isolation property of 1G as in (2) holds for G; if the real rank ofG is at least 2 or G is a rank one group of type Sp(m, 1) or F−20
4 , Ghas the so-called Kazhdan’s property (T) [29], which says that the trivialrepresentation of G is isolated in the whole unitary dual of G. When G isisomorphic to SO(m, 1) or SU(m, 1) which do not have Kazhdan’s property(T), the isolation of the trivial representation is still true in the subset ofall automorphic representations L2(Γ(q)\G)’s, due to the work of Selberg,Burger-Sarnak [11] and Clozel [14]. This latter property is referred as thephenomenon that G has property τ with respect to the congruence familyΓ(q).
Therefore, we have:
Theorem 5.2. If Γ is of finite index in G(Z), then the family Xq =C(Γ(q)\Γ, S) : q ∈ N is an expander family.
APOLLONIAN PACKINGS 21
In the case when Γ is of infinite index, the trivial representation is notcontained in L2(Γ(q)\G), as the constant function is not square-integrable,and the above correspondence cannot be used, and deciding whether Xq
forms an expander 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 Γk generated by Sk has finite index only for k = 1, 2 and hence weknow the family Xq(k) = C(Γk(q)\Γk, Sk) forms an expander for k = 1, 2by Theorem 5.2 but the subgroup Γ3 generated by S3 has infinite index inSL2(Z) and it was not known whether Xq(3) is an expander family untilthe work of Bourgain, Gamburd and Sarnak [3].
Theorem 5.3 (Bourgain-Gamburd-Sarnak [3], Salehi-Golsefidy-Varju [51]).Let Γ < G(Z) be a thin subgroup, i.e., Γ is Zariski dense in G. Let S be a fi-nite symmetric generating subset of Γ. Then C(Γ(q)\Γ, S) : q: square-freeforms an expander family.
If G is simply connected in addition, the strong approximation theoremof Matthews, Vaserstein and Weisfeiler [33] says that there is a finite set Bof primes such that for all q with no prime factors from B, Γ(q)\Γ is isomor-phic to the finite group G(Z/qZ) via the canonical projection Γ→ G(Z/qZ);hence the corresponding Cayley graph C(G(Z/qZ), S) is connected. Simi-larly, Theorem 5.3 says that the Cayley graphs C(G(Z/qZ), S) with q square-free and with no factors from B are highly connected, forming an expanderfamily; 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 byPyber-Szabo [47] and Breuillard-Green-Tao [9] (see also [25]).
The study of expanders has many surprising applications in various areasof mathematics (see [54]). We describe its application in sieves, i.e., inthe study of primes. For motivation, we begin by considering an integralpolynomial f ∈ Z[x]. The following is a basic question:
Are there infinitely many integers n ∈ Z such that f(n) is prime?
• If f(x) = x, the answer is yes; this is the infinitude of primes.• If f(x) = ax + b, the answer is yes if and only if a, b are co-prime.
This is Dirichlet’s theorem.• If f(x) = x(x+ 2), then there are no primes in f(Z) for an obvious
reason. On the other hand, Twin prime conjecture says that thereare infinitely many n’s such that f(n) is a product of at most 2primes. Indeed, Brun introduced what is called Brun’s combinato-rial sieve to attack this type of question, and proved that there areinfinitely many n’s such that f(n) = n(n + 2) is 20-almost prime,i.e., a product of at most 20 primes. His approach was improvedby Chen [12] who was able to show such a tantalizing theorem thatthere are infinitely many n’s such that f(n) = n(n + 2) is 3-almostprime.
22 HEE OH
In view of the last example, the correct question is formulated as follows:
Is there R <∞ such that the set of n ∈ Z such that f(n) is R-almostprime is infinite?
Bourgain, Gamburd and Sarnak [4] made a beautiful observation thatBrun’s combinatorial sieve can also be implemented for orbits of Γ on anaffine space via affine linear transformations and the expander property ofthe Cayley graphs of the congruence quotients of Γ provides a crucial inputneeded in executing the sieve machine.
Continuing our setup that G ⊂ SLN and Γ < G(Z), we consider the orbitO = v0Γ ⊂ ZN for a non-zero v0 ∈ ZN and let f ∈ Q[x1, · · · , xN ] such thatf(O) ⊂ Z.
Theorem 5.4 (Bourgain-Gamburd-Sarnak [4], Sarnak-Salehi-Golsefidy [52]).There exists R = R(O, f) ≥ 1 such that the set of vectors v ∈ O such thatf(v) is R-almost prime is Zariski dense in v0G.
We ask the following finer question:
Describe the distribution of the set v ∈ O : f(v) is R-almost prime in thevariety v0G.
In other words, is the set in concern focused in certain directions in O orequi-distributed in O? This is a very challenging question at least in the samegenerality as the above theorem, but when G is the orthogonal group of theDescartes quadratic form Q, Q(v0) = 0, and Γ is the integral Apolloniangroup, we are able to give more or less a satisfactory answer by [30] and[32]. More generally, we have the following: Let F be an integral quadraticform of signature (n, 1) and let Γ < SOF (Z) be a geometrically finite Zariskidense subgroup. Suppose that the critical exponent δ of Γ is bigger than(n− 1)/2 if n = 2, 3 and bigger than n− 2 if n ≥ 4. Let v0 ∈ Zn+1 be non-zero and O := v0Γ. We also assume that the skinning measure associatedto v0 and Γ is finite.
Theorem 5.5 (Mohammadi-O. [38]). Let f = f1 · · · fk ∈ Q[x1, · · · , xn+1]be a polynomial with each fi absolutely irreducible and distinct with rationalcoefficients and fi(O) ⊂ Z. Then we construct an explicit locally finitemeasure M on the variety v0G, depending on Γ such that for any familyBT of subsets in v0G which is effectively well-rounded with respect to M, wehave
(1) Upper bound: #v ∈ O∩BT : each fi(v) is prime M(BT )(logM(BT ))k
;
(2) Lower bound: Assuming further that maxx∈BT ‖x‖ M(BT )β
for some β > 0, there exists R = R(O, f) > 1 such that
#v ∈ O ∩ BT : f(v) is R-almost prime M(BT )
(logM(BT ))k.
The terminology of BT being effectively well-rounded with respect to Mmeans that there exists p > 0 such that for all small ε > 0 and for all
APOLLONIAN PACKINGS 23
T 1, the M-measure of the ε-neighborhood of the boundary of BT is atmost of order O(εpM(BT )) with the implied constant independent of ε andT . For instance, the norm balls v ∈ v0G : ‖v‖ ≤ T and many sectors areeffectively well-rounded (cf. [38]).
When Γ is of finite index, M is just a G-invariant measure on v0G andthis theorem 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 = v ∈ R4 : Q(v) = 0, ‖v‖max ≤ T is the max-norm ball, then forany primitive integral Apollonian packing P, the number of prime circles inP of curvature at most T is bounded by
4∑i=1
#v ∈ vPA ∩ BT , f(v) := vi prime
which is bounded by M(BT )log(M(BT )) by Theorem 5.5. Since we have M(BT ) =
c · Tα +O(Tα−η) where α = 1.305... is the critical exponent of A, this givesan upper bound Tα/ log T for the number of prime circles of curvature atmost T , as stated in Theorem 4.6. The upper bound for twin prime circlecount can be done similarly with f(v) = vivj .
Here are a few words on Brun’s combinatorial sieve and its use in Theorem5.5. Let A = am be a sequence of non-negative numbers and let B bea finite set of primes. For z > 1, let Pz =
∏p/∈B,p<z p and S(A, Pz) :=∑
(m,Pz)=1 am. To estimate Sz := S(A, Pz), we need to understand how A is
distributed along arithmetic progressions. For q square-free, define
Aq := am ∈ A : n ≡ 0(q)and set |Aq| :=
∑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
|Aq| = g(q)X + rq(A)
where g is a function on square-free integers with 0 ≤ g(p) < 1, gis multiplicative outside B, i.e., g(d1d2) = g(d1)g(d2) if d1 and d2
are square-free integers with (d1, d2) = 1 and (d1d2, B) = 1, and forsome c1 > 0, g(p) < 1− 1/c1 for all prime p /∈ B.
(A2) A has level distribution D, in the sense that for some ε > 0 andCε > 0, ∑
q<D
|rq(A)| ≤ CεX 1−ε.
(A3) A has sieve dimension k in the sense that there exists c2 > 0 suchthat for all 2 ≤ w ≤ z,
−c2 ≤∑
(p,B)=1,w≤p≤z
g(p) log p− r logz
w≤ c2.
24 HEE OH
Then for s > 9r, z = D1/s and X large enough,
S(A, Pz) X
(logX )k.
For our orbit O = v0Γ and f as in Theorem 5.5, we set
am(T ) := #x ∈ O ∩ BT : f(x) = m;Γv0(q) := γ ∈ Γ : v0γ ≡ v0 (q),|A(T )| :=
∑m
am(T ) = #O ∩ BT ;
|Aq(T )| :=∑
m≡0(q)
am(T ) = #x ∈ O ∩ BT : f(x) ≡ 0 (q).
Suppose we can verify the sieve axioms for these sequences Aq(T ) and zof order T η. Observe that if (f(v) = m,Pz) = 1, then all prime factors ofm have to be at least of order z = T η. It follows that if f(v) = m has R
prime factors, then T ηR m T degree(f), and hence R (degree f)/η.Therefore, Sz :=
∑(m,Pz)=1 am(T ) gives an estimate of the number of all
v ∈ O such that f(v) is R-almost prime for R = (degree f)/η.In order to verify these sieve axioms for O = v0Γ, we replace Γ by its
preimage under the spin cover G of G, so that Γ satisfies the strong approxi-mation property that Γ(q)\Γ = G(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.,
∑m=0 (q) am(T ) for all square-free integers
q, more precisely, we need to have a uniform control on the remainder termrq of Aq(T ) =
∑m=0 (q) am = g(q)X + rq such as rq X 1−ε for some ε > 0
independent of q. By writing
Aq(T ) =∑
γ∈Γv0 (q)\Γ,f(v0γ)=0 (q)
#(v0Γv0(q)γ ∩ BT )
the following uniform counting estimates provide such a control on the re-mainder term:
Theorem 5.7 (Mohammadi-O. [38]). Let Γ and BT be as in Theorem 5.5.For any γ ∈ Γ and any square-free integer q,
#v0Γ(q)γ ∩ BT =c0
[Γ : Γ(q)]M(BT ) +O(M(BT )1−ε)
where c0 > 0 and ε > 0 are independent over all γ ∈ Γ and q.
A basic ingredient of Theorem 5.7 is a uniform spectral gap for the Lapla-cian acting on L2(Γ(q)\Hn). Note that zero is no more the base-eigenvalueof the Laplacian when Γ(q) is a thin group, but δ(n−1−δ) is by Sullivan [57]and Lax-Phillips [31]. However, the expander result (Theorem 5.3) impliesa uniform lower bound for the gap between the base eigenvalue δ(n− 1− δ)and the next one; this transfer property was obtained by Bourgain, Gam-burd and Sarnak. As explained in section 3, the mixing of frame flow of the
APOLLONIAN PACKINGS 25
Bowen-Margulis-Sullivan measure is a crucial ingredient in obtaining themain term in Theorem 5.7, and the (uniform) error term in the countingstatement of Theorem 5.7 is again a consequence of a uniform error term inthe effective mixing of frame flow, at least under our hypothesis on δ.
References
[1] Jean Bourgain. Integral Apollonian circle packings and prime curvatures. Preprint.arXiv:1105:5127, 2011
[2] Jean Bourgain and Elena Fuchs. A proof of the positive density conjecture for integerApollonian packings J. Amer. Math. Soc. 24, 945–967, 2011
[3] Jean Bourgain, Alex Gamburd, and Peter Sarnak. Affine linear sieve, expanders, andsum-product Inventiones 179, (2010) 559-644
[4] Jean Bourgain, Alex Gamburd, and Peter Sarnak. Generalization of Selberg’s 3/16theorem and Affine sieve Acta Math. 207, (2011) 255–290
[5] Jean Bourgain and Alex Kontorovich. On the Local-Global conjecture for IntegralApollonian gaskets To appear in Inventiones. arXiv:1205:4416, 2012
[6] Jean Bourgain. Some Diophantine applications of the theory of group expansion InThin groups and superstrong approximation, edited by Breulliard and Oh, MSRI Publ61, Cambridge press.
[7] David W. Boyd. The sequence of radii of the Apollonian packing. Math. Comp.,39(159):249–254, 1982.
[8] Martine Babillot. On the mixing property for hyperbolic systems. Israel J. Math.,129:61–76, 2002.
[9] E. Breuillard, B. Green and T. Tao. Approximate subgroups of linear groups. Geom.Funct. Anal. 21 (2011) 774–819
[10] Marc Burger. Horocycle flow on geometrically finite surfaces. Duke Math. J.,61(3):779–803, 1990.
[11] Marc Burger and Peter Sarnak. Ramanujan duals II. Inventiones, 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] H. S. M. Coxeter. The problem of Apollonius. Amer. Math. Monthly, 75:5–15, 1968.[14] L. Clozel. Demonstration de la conjecture . Invent. Math. 151 (2003), pp. 297–328.[15] W. Duke, Z. Rudnick, and P. Sarnak. Density of integer points on affine homogeneous
varieties. Duke Math. J., 71(1):143–179, 1993.[16] Nicholas Eriksson and Jeffrey C. Lagarias. Apollonian circle packings: number theory.
II. Spherical and hyperbolic packings. Ramanujan J., 14(3):437–469, 2007.[17] Alex Eskin and C. T. McMullen. Mixing, counting, and equidistribution in Lie groups.
Duke Math. J., 71(1):181–209, 1993.[18] Elena Fuchs. Counting problems in Apollonian packings. Bulletin of AMS, Vol.50
(2013) 229-266[19] E. Fuchs. Strong Approximation in the Apollonian group. J. Number Theory, Vol
131, pp. 2282-2302 (2011)[20] Elena Fuchs and K. Sanden. Some experiments with integral Apollonian circle pack-
ings. Exp. Math. 20,380-399, 2011[21] L. Flaminio and R. Spatzier. Geometrically finite groups, Patterson-Sullivan measures
and Ratner’s theorem. Inventiones, 99, 601-626, 1990.[22] A. Gorodnik and A. Nevo. Lifting, restricting and sifting integral points on affine
homogeneous varieties. To appear in Compositio Math, 2012.[23] Ronald L. Graham, Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, and
Catherine H. Yan. Apollonian circle packings: geometry and group theory. I. TheApollonian group. Discrete Comput. Geom., 34(4):547–585, 2005.
26 HEE OH
[24] R.L. Graham, J.C. Lagarias, C.L. Mallows, A.R. Wilks, C.H. Yan. Apollonian circlepackings: number theory. J. Number Theory 100, pp. 1-45 (2003)
[25] B. Green. Approximate groups and their applications:work of Bourgain, Gamburd,Helfgott and Sarnak. Current Events Bulletin of the AMS, 2010
[26] H. Halberstam and H. Richert. Sieve methods. Academic Press., (1974) 167–242.[27] Harald Helfgott. Growth and generation in SL2(Z/pZ) Ann. Math., 167. 601–623,
2008.[28] Roger Howe and Calvin Moore. Asymptotic properties of unitary representations J.
Funct. Anal., 72–96, 1979.[29] David Kazhdan. On a connection between the dual space of a group and the structure
of its closed subgroups Func. Anal. Appl. 1 (1967), 63–65.[30] Alex Kontorovich and Hee Oh. Apollonian circle packings and closed horospheres on
hyperbolic 3-manifolds. Journal of AMS, Vol 24. 603–648, 2011.[31] Peter D. Lax and Ralph S. Phillips. The asymptotic distribution of lattice points in
Euclidean and non-Euclidean spaces. J. Funct. Anal., 46(3):280–350, 1982.[32] Min Lee and Hee Oh. Effective circle count for Apollonian packings and closed horo-
spheres. GAFA. Vol 23 (2013), 580–621[33] C. Matthews, L. Vaserstein and B. Weisfeiler. Congruence properties of Zariski dense
subgroups. Proc. London Math. Soc. 48, 1984, 514-532.[34] Gregory Margulis. On some aspects of the theory of Anosov systems. Springer Mono-
graphs in Mathematics. Springer-Verlag, Berlin, 2004. With a survey by RichardSharp: Periodic orbits of hyperbolic flows, Translated from the Russian by ValentinaVladimirovna Szulikowska.
[35] Gregory Margulis. Explicit constructions of expanders. Problems of InformationTransmission. 9 (1973), no. 4, 325-332.
[36] Mattila, Pertti. Geometry of sets and measures in Euclidean spaces. Cambridge Uni-versity Press. ISBN 978-0-521-65595-8.
[37] C. T. McMullen. Hausdorff dimension and conformal dynamics. III. Computation ofdimension. Amer. J. Math., 120(4):691–721, 1998.
[38] Amir Mohammadi and Hee Oh. Matrix coefficients, Counting and Primes for geo-metrically finite groups. To appear in J. EMS, arXiv 1208.4139
[39] D. Mauldin and M. Urbanski. Dimension and measures for a curvilinear Sierpinskigasket or Apollonian packings. Adv. Math., 136 (1998), 26-38
[40] David Mumford, Caroline Series, and David Wright. Indra’s pearls. Cambridge Uni-versity Press, New York, 2002.
[41] Amos Nevo and Peter Sarnak. Prime and Almost prime integral points on principalhomogeneous spaces Acta Math, 205 (2010), 361–402
[42] Hee Oh. Dynamics on Geometrically finite hyperbolic manifolds with applications toApollonian circle packings and beyond. Proc. of ICM. (Hyderabad, 2010), Vol III1308–1331
[43] Hee Oh. Harmonic Analysis, Ergodic theory and Counting for Thin groups. In Thingroups and superstrong approximation, edited by Breulliard and Oh, MSRI Publ 61,Cambridge press.
[44] Hee Oh and Nimish Shah. Equidistribution and counting for orbits of geometricallyfinite hyperbolic groups. Journal of AMS. Vol 26 (2013) 511–562
[45] Hee Oh and Nimish Shah. The asymptotic distribution of circles in the orbits ofKleinian groups. Inventiones, Vol 187, 1–35, 2012
[46] S.J. Patterson. The limit set of a Fuchsian group. Acta Mathematica, 136:241–273,1976.
[47] L. Pyber, E. Szabo. Growth in finite simple groups of Lie type of bounded rankPreprint (2011) arXiv:1005.1858.
[48] J. G. Ratcliffe. Foundations of hyperbolic manifolds. Springer-Verlag, GTM 149
APOLLONIAN PACKINGS 27
[49] Thomas Roblin. Ergodicite et equidistribution en courbure negative. Mem. Soc. Math.Fr. (N.S.), (95):vi+96, 2003.
[50] Daniel Rudolph. Ergodic behavior of Sullivan’s geometric measure on a geometricallyfinite hyperbolic manifold. Erg. Th. and Dyn. Syst., Vol 2, 1982, 491–512
[51] A. Salehi Golsefidy and P. Sarnak. Affine Sieve To appear in JAMS. arXiv:1109.6432[52] A. Salehi Golsefidy and P. Varju. Expansion in perfect groups GAFA Vol 22 (2012),
1832–1891[53] Peter Sarnak. Integral Apollonian packings. Amer. Math. Monthly. 118 (2011), 291-
306[54] Peter Sarnak. Notes on Thin matrix groups In Thin groups and superstrong approx-
imation, edited by Breulliard and Oh, MSRI Publ 61, Cambridge press.[55] B. Stratmann and M. Urbanski. The box-counting dimension for geometrically finite
Kleinian groups Fund. Math., (149):83–93, 1996.[56] Dennis Sullivan. The density at infinity of a discrete group of hyperbolic motions.
Inst. Hautes Etudes Sci. Publ. Math., (50):171–202, 1979.[57] Dennis Sullivan. Entropy, Hausdorff measures old and new, and limit sets of geomet-
rically finite Kleinian groups. Acta Math., 153(3-4):259–277, 1984.[58] Ilya Vinogradov. Effective bisector estimate with applications to Apollonian circle
packings. preprint, 2012, arXiv.1204.5498[59] Dale Winter. Mixing of frame flow for geometrically finite rank one manifold with
application to measure classification. preprint, 2013.
Mathematics department, Yale university, New Haven, CT 06520 and KoreaInstitute for Advanced Study, Seoul, Korea
E-mail address: [email protected]
