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\n An Apollonian circle packing (ACP) is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In this paper, we compute a lower bound for the number\n \n \n \n \n \n \u03ba\n \n <\/mml:mi>\n (<\/mml:mo>\n P<\/mml:mi>\n ,<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\kappa (P,X)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of integers less than\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n occurring as curvatures in a bounded integer ACP\n \n \n \n P<\/mml:mi>\n P<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and prove a conjecture of Graham, Lagarias, Mallows, Wilkes, and Yan that the ratio\n \n \n \n \n \n \u03ba\n \n <\/mml:mi>\n (<\/mml:mo>\n P<\/mml:mi>\n ,<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n X<\/mml:mi>\n <\/mml:mrow>\n \\kappa (P,X)\/X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is greater than\n \n \n \n 0<\/mml:mn>\n 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n tending to infinity.\n <\/p>","DOI":"10.1090\/s0894-0347-2011-00707-8","type":"journal-article","created":{"date-parts":[[2011,6,20]],"date-time":"2011-06-20T17:02:36Z","timestamp":1308589356000},"page":"945-967","source":"Crossref","is-referenced-by-count":36,"title":["A proof of the positive density conjecture for integer Apollonian circle packings"],"prefix":"10.1090","volume":"24","author":[{"given":"Jean","family":"Bourgain","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elena","family":"Fuchs","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2011,6,20]]},"reference":[{"key":"1","unstructured":"[Be] P. 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Amer. Math. Soc."],"abstract":"

Nearly twenty years ago, a paper of Graham, Lagarias, Mallows, Wilks, and Yan on the number theory of Apollonian circle packings sparked an interest in the number theory community, which was just developing tools to handle arithmetic problems involving so-called thin groups. At the time, these packings were the only naturally occurring example of such an arithmetic problem and, naturally, number theorists sprang upon the opportunity to discover all their rich properties, thinness notwithstanding. In 2010, building upon conjectures of Graham et al., the author and Katherine Sanden gave evidence toward the local to global conjecture for Apollonian circle packings, stating that in any integral packing, any large enough integer that satisfied certain congruence conditions modulo 24 must appear as a curvature in the packing. For thirteen years, most everyone believed this conjecture to be true. In this article, we will explore the history of this conjecture and its fascinating downfall after Haag, Kertzer, Rickards, and Stange proved that, in fact, infinitely many integral Apollonian packings fail to abide by the local to global principle, and come with extra quadratic and quartic obstructions.<\/p>","DOI":"10.1090\/bull\/1874","type":"journal-article","created":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T14:57:58Z","timestamp":1767797878000},"page":"59-77","source":"Crossref","is-referenced-by-count":0,"title":["Apollonian packings: The rise and fall of the local to global conjecture"],"prefix":"10.1090","volume":"63","author":[{"given":"Elena","family":"Fuchs","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2025,10,15]]},"reference":[{"issue":"3","key":"1","first-page":"104","article-title":"Geometric sequences of discs in the Apollonian packing","volume":"9","author":"Aharonov, D.","year":"1997","journal-title":"Algebra i Analiz","ISSN":"https:\/\/id.crossref.org\/issn\/0234-0852","issn-type":"print"},{"key":"2","unstructured":"[2] L. 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Amer. Math. Soc."],"abstract":"

\n An Apollonian circle packing is a classical construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature, making the packings of interest from a number theoretic point of view. Many of the natural arithmetic problems have required new and sophisticated tools to solve them. The reason for this difficulty is that the study of Apollonian packings reduces to the study of a subgroup of\n \n \n \n \n \n \n GL<\/mml:mtext>\n <\/mml:mrow>\n 4<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n \n Z<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \\textrm {GL}_4(\\mathbb Z)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n that is thin in a sense that we describe in this article, and arithmetic problems involving thin groups have only recently become approachable in broad generality. In this article, we report on what is currently known about Apollonian packings in which all circles have integer curvature and how these results are obtained. This survey is also meant to illustrate how to treat arithmetic problems related to other thin groups.\n <\/p>","DOI":"10.1090\/s0273-0979-2013-01401-0","type":"journal-article","created":{"date-parts":[[2013,1,16]],"date-time":"2013-01-16T14:59:47Z","timestamp":1358348387000},"page":"229-266","source":"Crossref","is-referenced-by-count":17,"title":["Counting problems in Apollonian packings"],"prefix":"10.1090","volume":"50","author":[{"given":"Elena","family":"Fuchs","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2013,2,14]]},"reference":[{"issue":"3","key":"1","first-page":"104","article-title":"Geometric sequences of discs in the Apollonian packing","volume":"9","author":"Aharonov, D.","year":"1997","journal-title":"Algebra i Analiz","ISSN":"https:\/\/id.crossref.org\/issn\/0234-0852","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/BF02579166","article-title":"Eigenvalues and expanders","volume":"6","author":"Alon, N.","year":"1986","journal-title":"Combinatorica","ISSN":"https:\/\/id.crossref.org\/issn\/0209-9683","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/0095-8956(85)90092-9","article-title":"\ud835\udf06\u2081, isoperimetric inequalities for graphs, and superconcentrators","volume":"38","author":"Alon, N.","year":"1985","journal-title":"J. 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