Arithmetic Properties of Apollonian Circle Packings

85 Pages·2010·0.58 MB·English

Arithmetic Properties of Apollonian Circle Packings
Elena Fuchs
ABSTRACT. An Apollonian circle packing (ACP) is an ancient Greek construction which is made by
repeatedlyinscribingcirclesintothetriangularintersticesinaDescartesconfigurationoffourmutually
tangentcircles. Remarkably, iftheoriginalfourcircleshaveintegercurvature, allofthecirclesinthe
packingwillhaveintegercurvatureaswell,makingthepackingsofgreatinterestfromanumbertheoretic
pointofview. ThispointofviewhasbeenexploredextensivelybyGraham,Lagarias,Mallows,Wilkes,
andYanaspartofaseriesofpapersonACP’s.Inthisthesis,weusethecorrespondencebetweeninteger
ACP’sandorbitsofacertainSchottkygroupcombinedwiththerecentlydevelopedaffinelinearsieveof
Bourgain,Gamburd,andSarnaktoanswermanyofthequestionsraisedbyGrahamet.al.intheiraccount.
Acknowledgements
There are many people without whose help and support this thesis would not have been possible.
Firstofall,Iwouldliketothankmyadviser,PeterSarnak,forintroducingmetothebeautifullysimple
problem of Apollonian circle packings and the not so simple affine sieve. He has taught me many
important lessons not only about mathematics itself, but also about how to do mathematics and will
always remain an inspiration and role model to me as I continue my career in mathematics. Besides
my adviser, I heartily thank many of the students, faculty, and staff of the Princeton mathematics
departmentfortheiradviceandsupport,bothmathematicalandotherwise,duringmytimeasagraduate
student.
IwouldalsoliketothankAlexKontorovich,JeffLagarias,AlirezaSalehi-Golsefidy,andKatherine
Sanden for many insightful suggestions and conversations about the work in this thesis. Great thanks
tomyco-authorofthefourthchapterofthisthesis,JeanBourgain.
A warm thank you as always to my wonderful friends and my family Irina, Dmitry, and Katia for
theirloveandsupportoverthelastfiveyears.
3

Contents
Acknowledgements 3
TheKissPrecise 7
Chapter1. Introduction 9
1. Historyandsetup 9
2. TheApolloniangroup 15
Chapter2. CongruenceObstructions 19
1. ThepreimageΓofAinSL (C) 20
2
2. ThereductionofΓmodulosquarefree(d) 21
3. ThereductionofΓmoduloanyideal(d) 26
4. Congruenceobstructionsfortheorbit 29
Chapter3. Sieving 37
1. Theaffinesieveandtheimportanceofexpanders 37
2. Badprimes 40
3. 28-almostprimepoints 44
4. Aprimenumbertheorem 50
Chapter4. DensityofCurvatures(jointwithJeanBourgain) 65
1. Apreliminarylowerbound 67
2. Countinginseveralsubpackingsatonce 70
Chapter5. Appendix 81
Bibliography 83
5

The Kiss Precise
Forpairsoflipstokissmaybe
Involvesnotrigonometry.
’Tisnotsowhenfourcircleskiss
Eachonetheotherthree.
Tobringthisoffthefourmustbe
Asthreeinoneoroneinthree.
Ifoneinthree,beyondadoubt
Eachgetsthreekissesfromwithout.
Ifthreeinone,thenisthatone
Thricekissedinternally.
Fourcirclestothekissingcome.
Thesmallerarethebenter.
Thebendisjusttheinverseof
Thedistancefromthecenter.
ThoughtheirintrigueleftEucliddumb
There’snownoneedforruleofthumb.
Sincezerobend’sadeadstraightline
Andconcavebendshaveminussign,
Thesumofthesquaresofallfourbends
Ishalfthesquareoftheirsum.
Tospyoutsphericalaffairs
Anoscularsurveyor
Mightfindthetasklaborious,
Thesphereismuchthegayer,
Andnowbesidesthepairofpairs
Afifthsphereinthekissingshares.
Yet,signsandzeroasbefore,
Foreachtokisstheotherfour
Thesquareofthesumofallfivebends
Isthricethesumoftheirsquares.
–FrederickSoddy,1936
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CHAPTER 1
Introduction
OneofthemostessentialtoolsinnumbertheoryisthetheoryofautomorphicformsandL-functions
(see [45] and [31] for an elegant survey of various applications). For example, several long-standing
problemsinanalyticnumbertheoryhavebeenreducedtofindinggoodestimatesforFouriercoefficients
of automorphic forms, and so, with or without applications in mind, automorphic forms have been
studiedextensivelythroughoutthepastcenturyandarestillofgreatinteresttoday.
One major aspect of these tools is understanding the spectral theory of the Laplace operator ∆ on
L2(Γ,H), where Γ is classically taken to be a congruence subgroup of SL (R) (more generally one
2
might consider automorphic forms on GL over a number field). For example, it is known that the
n
baseeigenvalueinthespectrumforsuchgroupsisλ =0,correspondingtotheconstanteigenfunction.
0
In the case Γ = Γ (N), the modular group, Selberg’s eigenvalue conjecture states that there are no
0
eigenvalues0<λ <1/4,andthereareanalogsofthisconjectureinmoregeneralcasesaswell.
However,inthecasethatΓ\Hhasinfinitevolume(wecallΓthinifthisisthecase)itisunclearhow
thesetoolswouldapplyandmuchlessisknown. Forexample,itisnolongertruethatthebaseeigen-
valueis0–infact,theconstantfunctionisnolongersquareintegrableinthissituation! Nevertheless,
there is a wealth of diophantine problems which can be rephrased precisely in terms of such a group
Γ. In this thesis, we approach a particularly beautiful example of this kind of problem and are able
to solve several relevant open questions without relying on automorphic forms – rather, certain rich
propertiesofthegroupinquestionallowustomakeuseofthearithmeticanalysisrecentlycarriedout
byBourgain,Gamburd,andSarnakin[7]. Thisexampleismeanttoconvincethereaderthatitisvery
naturaltoconsiderdiophantineproblemsassociatedwiththingroupsaswellastooutlinethemethods
onemightusetoaddressthem.
1. Historyandsetup
Consider four mutually tangent circles, one of them internally tangent to the other three as in the
first picture in Fig. 1. One might ask whether there is a unique way to inscribe a circle into each of
the curvilinear triangles in this picture. In fact, this uniqueness follows from an ancient theorem of
ApolloniusofPerga:
THEOREM1.1. (Apollonius,circa200BC):Toanythreemutuallytangentcirclesorlinesthereare
preciselytwoothercirclesorlineswhicharetangenttoallthree.
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10 1.INTRODUCTION
Apolloniusdiscoveredthiswhileattemptingthedifficulttaskofconstructingmutuallytangentcircles
and straight lines using only a straight edge and compass. Coming back to our picture in Fig. 1,
Theorem1.1impliesthatthereisauniquecirclewhichcanbeinscribedintoeverycurvilineartriangle
produced by the four mutually tangent circles we have constructed (see the second picture in Fig. 1).
We now have 12 new curvilinear triangles, each of which can be filled with a unique inscribed circle
again. Thisprocesscanbecontinuedindefinitely,andtheresultingpictureisdulycalledanApollonian
circle packing (ACP). Given this process of constructing the packing, we say that the original four
circlesinthefirstpictureofFig.1areborninthefirstgenerationofthepacking,thenewcirclesinthe
secondpictureareborninthesecondgeneration,andsoon.

! Generation 1 Generation 2 Generation 3
FIGURE 1. PackingCircles
One can study Apollonian circle packings from many different angles – various properties of the
packings are investigated in a beautiful series of papers by Graham, Lagarias, Mallows, Wilkes, and
Yan (see [24], [21], [22], [23]). Such packings are certainly of interest in classical geometry – for
example,theprocessofproducinganewcircleatgenerationkisinfactequivalenttoreflectingoneof
thecirclesbornatgenerationk−1inanothercircleorstraightline(seeSection2.1foradiscussionof
this).
We are particularly interested in the number-theoretic questions about ACP’s which are addressed
extensively by Graham et.al. in [24]. To understand how these questions arise in the context of this
purely geometric construction, consider the curvatures, or reciprocals of the radii, of the circles in
a given ACP (equivalently one may consider the radii but these quickly tend to zero, so it is more
convenienttoworkwiththecurvaturesinstead). ThenumbertheoreticaspectofACP’stracesbackto
the following theorem due to Descartes, that the curvatures of any four mutually tangent circles in an
ACPsatisfyaquadraticequation:
THEOREM 1.2. (Descartes,1643): Leta,b,c,andd denotethecurvaturesoffourmutuallytangent
circles,whereacirclehasnegativecurvatureiffitisinternallytangenttotheotherthree. Then
(1.1) Q(a,b,c,d):=2(a2+b2+c2+d2)−(a+b+c+d)2=0.