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Apollonian Circle Packings: Dynamics and Number Theory

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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 7 3. Counting, Mixing, and the Bowen-Margulis-Sullivan measure 9 4. Integral Apollonian circle packings 15 5. Expanders and Sieve 19 References 25 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 2 HEE OH Figure 2. Possible configurations of four mutually tangent circles Proof. We give a modern proof, using the linear fractional transformations ^ of PSL2(C) on the extended complex plane C = C [ f1g, known as M¨obius transformations: a b az + b (z) = ; c d cz + d where a; b; c; d 2 C with ad − bc = 1 and z 2 C [ f1g. As is well known, a M¨obiustransformation maps circles in C^ to circles in C^, preserving angles between 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 by p the tangent point between C1 and C2, and let g 2 PSL2(C) be an element which maps p to 1. Then g maps C1 and C2 to two circles tangent at 1, 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 there 0 are precisely two circles, say, D and D tangent to all three g(Ci), 1 ≤ i ≤ 3. Using g−1, which is again a M¨obiustransformation, it follows that g−1(D) −1 0 and g (D ) are precisely those two circles tangent to C1,C2,C3. In order to construct an Apollonian circle packing, we begin with four mutually tangent circles in the plane (see Figure 2 for possible configura- tions) and keep adding newer circles tangent to three of the previous circles provided by Theorem 1.1. Continuing this process indefinitely, we arrive at an infinite circle packing, called an Apollonian circle packing. APOLLONIAN PACKINGS 3 38 27 27 86 131 95 54 54 110 18 18 167 155 182 63 63 138 74 134 74 195 231 170 11 30 11 11 266 30 59 95 95 251 191 59 194 98 183 98 98 294 314 215 267 110 110 39 39 299 242 254 83 83 143 107 66 66 194 215 234 35 179 290 35 90 314 279 90 111 111 111350 111 6 6 6 6 338 6 227 171 455 174 59 423 59 174 506 347 162 162486 315 255 131 354 354 131 179 179471 378 522 299 371 150 47 98 294 194 167 98 47 62 507 62 363 23 119 23 23 362 119 290 147 147 170 438 170447 14 14 375 330 330 14 75 75 122 122231 231 443 186 150 150302 50 50 383 275 251 182 87 87 87 278 87 26 134 218 26 131 2 2 2 2 143 42 42 62 167 114 371 114 374 546 219 219 71 447 71 654 554 222 659 222 531 455 279 738 279 846 623 210 210 347 911 38 38 107 662 107 38 318 842 318 635 546 287 867 287 323 642 323927 762 338 858 110 203 599 203 110 647 699 683 218 638 218 263 498 263 102 798 102 362 407 803 863 302 198 491 198 302 602 326 590 99 143 99 63 326 63 590 35 35 35 602 302 198 491 198 302 863 407 803 362 798 263 498 263 218 638 218 15 15 15 15 699 15 683 102 102 647 203 599 203 338 858 323 323 642 762 110 927 110 287 867 287 546 635 318 842 318 347 662 210 210 911 623 107 107846 279 738 279 455 38 38 531 38 222 222 659 554 654 219 447 219 71 546 71 374 371 114 114 167 3 3 3 3 3 3 3 62 42 143 42 131 134 218 87 87 87 278 182 87 26 251 26 275 383 150 150302 50 50 443 186 231 231 330 330 375 75 122 170 438 170447 122 75 147 147 362 290 363 507 194 167 98 294 98 150 119 371119 47 62 62 299 47 522 179 179471 378 23 131 23 23 354 354 131 255 315 162 162486 506 347 14 14 423 14 174 174 59 59 455 338 227 171 350 314 279 90 111 111 111 111 90 290 35 179 35 234 215 66 66 194 107 143 83 83 254 6 6 6 6 242 6 39 39 299 267 110 314 110215 98 98 294 183 194 251 191 98 59 95 95 59 266 30 30 170 231 74 134 74 195 63 63 138 2 2 2 11 11 2 155 182 11 167 110 54 95 54 18 18 131 27 27 86 38 Figure 3. First few generations 169 144 144 169 196 169 225 324 289 169 196 361 441 169324 289144361 256 256 361144289 324169441 361 196 169 289 324 225 169 196 169 144 144 169 516 516 81 121 100 121 81 81 121 100 121 81 121 81 121372 100 121388 436 81121 12181 436 388121 100 121372 81 121 81 121 100 121 81 81 121 100 121 81 316 316 268 268 64 64 64 64 244 64561 64561 244 64 64 64 64 49 49 49 49 49 49 49 49 49 49 513 49 49 513 49 49 49 49 49 49 49 49 49 49 196 196 196 456 456 196 196 196 417 417 36 156 156 36 36 396 396 156393 393 156 396 396 36 36 156 156 36 148 148 148 148 148 148 321 321 297 297 25 25 25 25 25 25 25 25 25 25 12425249 25 25 24925124 25 25 25 25 25 25 25 25 25 25 424 424 216 216 216625 625 216 216 216 600 384 384 600 201 201 204 577 201 201 577 204 201 201 84 84 84 537 537 84 84 84 76 76 76 76 177 76 76 177 76 76 76 76 304 472 472 304 16 16 16 16 16 16 16 457 457 16 16 16 16 16 16 16 153 153 153 153 433 153 153 153 153433 153 153 153 153 412 256 256 412 393 393 240 240 129 129 129348 348129 129 129 52 52 52 52 52337 337 52 52 52 52 52 313 313 108 108 108184 184108 108 108 265 265 160 160 496 160 160496 160 160 481 481 9 9 9 9 9 9 9 9 9 616 616 9 9 9 9 9 9 9 9 9 232 232 457232 232457 232 232 81 81 81 81 673 81 81673 81 81 81 81 576 576 220 568 436 648 648 436 568 220 136 217 217 217 217 136 136433 217 217 217 217 433136 136 217 217 217 217 136 625 417 417 625 72 72 72 72 409 72 72409 72 72 72 72 585 585 193 193 504 193 193504 193 193 372 372 28 28 28 28 28 28 28361 36128 28 28 28 28 28 28 177 177 520 177 177520 177 177 448 448 168 168 337168505 505 168337 168 168 424 424 57 57 57 57 57313 31357 57 57 57 57 96 96 304 96 96304 96 96 145 145 156 156 4 4 4 4 4 345 4 345 4 4 4 4 4 105 105 105336 505 505 105336 105 105 201 201 201412 412201 201 201 592 592 489 489 256 256 256681 681 256 256 256 64 64 64 64 64 576 576 64 64 64 64 64 192 193 193 192 481192577 768 193321 321193 768 577192481 192 193 193 192 600 600 97 97 97 97 388 808 97 97808 388 97 97 97 97 280 280 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324121 121 121 100 12164 64121 100 121 64 64 Figure 4.
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