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Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting PDF

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Springer Monographs in Mathematics József Beck Probabilistic Diophantine Approximation Randomness in Lattice Point Counting Springer Monographs in Mathematics Moreinformationaboutthisseriesathttp://www.springer.com/series/3733 József Beck Probabilistic Diophantine Approximation Randomness in Lattice Point Counting 123 JózsefBeck DepartmentofMathematics RutgersUniversity Piscataway,NJ,USA ISSN1439-7382 ISSN2196-9922(electronic) ISBN978-3-319-10740-0 ISBN978-3-319-10741-7(eBook) DOI10.1007/978-3-319-10741-7 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014950069 ©SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface p Wecouldchooserandomnessof 2asanalternativesubtitleofthebook.Indeed, p thebookconnectstwoseeminglyunrelatedconcepts,namely,(1) 2:symbolizing the class of quadratic irrationals, including the theory of the quadratic number fields in general and (2) randomness. These two concepts, representing algebra (thescienceoforderandstructure)andprobabilitytheory(thescienceofdisorder), are the endpointsof a long chain of relations/implications. The periodicity of the p continued fraction of 2 (or any other quadratic irrational) means self-similarity. Self-similarity leads to independence (e.g., via Markov chains; here we refer to the well known probabilistic concept), and independence ensures (nearly) perfect randomness.Inparticular,weprovesomeunexpectedprobabilisticresults: quadraticirrationalH)periodiccontinuedfractionH) H)self-similarityH)independence.orindependenceviaMarkovchains/H) H)randomnessWcentrallimittheoremandthelawoftheiteratedlogarithm Thisdiagrammaysummarizethebookinanutshell. p Thereasonwhywedecidednottochooserandomnessof 2tobethesubtitle is that it would perhaps mislead the reader. The reader would probably expect us to prove the apparent randomness of the digit distribution in the usual decimal expansion p 2D1:414213562373095048801688724209698078569671875376948:::: Unfortunately, we cannot make any progress with this famous old problem; it remains open and hopeless (to read more about this and other related famous open problems the reader may jump ahead right now to Sect.2.5: A Giant Leap in number theory). What we study instead is the “irrational rotation” by any v vi Preface p quadratic irrational, say, by 2. We study the global and local behavior of the irrationalrotationfromaprobabilisticviewpoint—thisexplainsthetitleofthebook probabilisticdiophantineapproximation. Consider the linear sequence n˛, n D 1;2;3;:::: it is perfectly regular, it is an infinite arithmetic progression. Even if we take it modulo one, and ˛ is an arbitrary (but fixed) irrational, the sequence n˛ (mod 1)—called irrational rotation—still features a lot of regularities. For example, (1) we have infinitely manyBoundedErrorIntervals,(2)wehaveinfinitelymanyBoundedErrorInitial Segments,(3)everyinitialsegmenthasatmostthreedifferent“gaps,”and(4)there isanextremelystrongrestrictionontheinducedpermutations—theseareallstrong “anti-randomness”typeregularitypropertiesof theirrationalrotationn˛ (mod1), n D 1;2;3;:::(properties(1)–(4)willbe explainedin depthsin Sect.1.1).These regularitiesshowthattheirrationalrotationishighlynon-randominmanyrespects. Thisiswhytheirrationalrotation(withanunderlyingnestedstructure)isalsocalled aquasi-periodicsequence. Also we know from number theory that the key to understand the irrational rotation n˛ (mod 1), n D 1;2;3;:::; is to know the continued fraction for ˛. The quadratic irrationals have the most regular continued fraction: the class of quadratic irrationals is characterized by the property of (ultimately) periodic continuedfraction,forexample, p 1 2D1C DŒ1I2;2;2;:::(cid:2)DŒ1I2(cid:2): 2C 1 2C(cid:2)(cid:2)(cid:2) Despitetheseregularitiesoftheirrationalrotation,ourfirstmainresultexhibits “full-blownrandomness.”Forexample,howmuchtimedoestheirrationalrotation n˛ (mod 1), n D 1;2;3;:::; spend in the first half Œ0;1=2/ of the unit interval Œ0;1/?Well,weproveacentrallimittheoremforeveryquadraticirrational˛ (e.g., p ˛ D 2). More precisely, let ˛ be an arbitrary real root of a quadratic equation p with integer coefficients, say, ˛ D 2. Given any rational number 0 < x < 1 (say,x D 1=2)andanypositiveintegern,wecountthenumberofelementsofthe sequence˛;2˛;3˛;:::;n˛ modulo1thatfallintothesubintervalŒ0;x(cid:2).Weprove that this counting number satisfies a central limit theorem in the following sense. First, we subtractthe “expectednumber"nx fromthe countingnumberand study the typical fluctuation of this difference as n runs in a long interval 1 (cid:2) n (cid:2) N. Dependingon˛ andx,wemayneedanextraadditivecorrectionofconstanttimes logarithm of N; furthermore,what we always need is a multiplicative correction: divisionby(another)constanttimessquarerootoflogarithmofN.IfN islarge,the distributionofthisrenormalizedcountingnumber,asnrunsin1(cid:2)n (cid:2)N,isvery closetothestandardnormaldistribution(bell-shapedcurve),andthecorresponding error term tends to zero as N tends to infinity. This is one of the main results of thebook(seeTheorem1.1).Theproofisrathercomplicatedandlong;ithasmany interesting detours and by-products. For example, the exact determination of the Preface vii keyconstantfactors(intheadditiveandmultiplicativenorming),whichdependon ˛andx,requiressurprisinglydeepalgebraictoolssuchasDedekingsums,theclass numberofquadraticfields,andgeneralizedclassnumberformulas. p Perhaps the reader is wondering: why are the quadratic irrationals (like 2) specialandworthspendinghundredsofpageson.Theansweristhatthequadratic irrationals play a central role in diophantine approximation for several reasons. Theyarethe“mostanti-rationalrealnumbers”(officiallycalledbadlyapproximable numbers), and at the same time they represent the most uniformly distributed irrationalrotations.AthirdreasonisthePell’sequationx2(cid:3)dy2 D ˙1(d (cid:4) 2is p squarefree),whichisofcoursecloselyrelatedto d.Also,andthisisthemessage of our book, the best way to understand the local and global randomness of the irrationalrotation is to focuson the class of quadraticirrationals.This class gives themostelegantandstrikingresultswiththesimplestproofs.Someoftheseresults extendtoalmosteveryrealnumber,someofthemdonotextend.Wewillelaborate oneachoneoftheseissueslater. The quadratic irrational rotation demonstrates the coexistence of order and p randomness; a novelty here is the much smaller norming factor logn (instead p of the usual n). The logn comes from the fact that the underlying problem is about“generalizeddigitsums”withthesurprisingtwistthatthebaseofthenumber p systemisanirrationalnumber(namely,thefundamentalunit,e.g.,itis1C 2for p ˛ D 2).Alsolognrepresentstheminimum;itcorrespondstothemostuniformly distributedirrationalrotations. OursecondmainsubjectismotivatedbytheclassicalPell’sequation.Findingthe integralsolutionsof(say)x2 (cid:3)2y2 D ˙1meanscountinglatticepointsinalong and narrowtilted hyperbolicregionthatwe call a “hyperbolicneedle.” Of course, we basically knoweverythingaboutPell’s equation(this is why Pell’s equationis included in every undergraduate number theory course), but what happens if we translatethe“hyperbolicneedle”?What istheasymptoticnumberoflattice points inside (note that the area is infinite)? Well, for a typical translated copy of the “hyperbolicneedle”—whichcorrespondstoan“inhomogeneousPellinequality”— weprovea“lawoftheiteratedlogarithm,”whichdescribestheasymptoticnumber ofintegralsolutionsinastrikinglypreciseway.Inotherwords,theclassicalCircle Problem of Gauss is wide open, but here we can solve an analogous Hyperbola Problem. This result is a good illustration of the full power of the probabilistic viewpoint in number theory. In general, consider the inhomogeneousdiophantine inequality c kn˛(cid:3)ˇk< ; (0.1) n where ˛ is an arbitrary irrational, ˇ, c > 0 are arbitrary real numbers, and n is the variable. An old result of Kronecker states that inequality (0.1) has infinitely manyintegralsolutionsnifc D3;thisishowKroneckerprovedthattheirrational viii Preface rotationn˛(mod1)isdenseintheunitinterval.Whatcanwesayaboutthenumber p ofsolutionsnofinequality(0.1)?Considerthespecialcase˛ D 2of(0.1): p c kn 2(cid:3)ˇk< ; (0.2) n p andletF. 2IˇIcIN/denotethenumberofintegralsolutionsnofinequality(0.2) satisfying 1 (cid:2) n (cid:2) N; this counting function is about the local behavior of the p p irrationalrotationn 2(mod1).WecandescribethetrueorderofF. 2IˇIcIN/, as N ! 1, in an extremely precise way for almost every ˇ. We prove that the p number of solutions F. 2IˇIcIen/ of (0.2) oscillates between the sharp bounds (">0) p p p p p 2cn(cid:3)(cid:3) n .2C"/loglogn<F. 2IˇIcIen/<2cnC(cid:3) n .2C"/loglogn (0.3) asn ! 1foralmosteveryˇ;seeTheorem5.6inPart1.3ofthebook.Notethat p (cid:3) D(cid:3). 2;c/>0isapositiveconstant,and (0.3)failswith2(cid:3)"insteadof2C". (Thereasonwhyin(0.3)weswitchedfromN totheexponentiallysparsesequence p enisthatthecountingfunctionF. 2IˇIcIN/isslowlychanginginthesensethat, p as N runs in en < N < enC1, F. 2IˇIcIN/ makes only an additive constant change.) Observethatinequality(0.2)is(basically)equivalenttotheinhomogeneousPell inequality (cid:3)c0 (cid:2).xCˇ/2(cid:3)2y2 (cid:2)c0; (0.4) p wherec0 D 2 2c. Notice thatequation(0.4)determinesa longandnarrowtilted hyperbolaregion(“hyperbolicneedle”).Themessageof(0.3)is,roughlyspeaking, that for almost all translations, the number of lattice points in long and narrow hyperbolasegmentsof any fixed quadraticirrationalslope equalsthe area plusan errortermwhichisnevermuchlargerthanthesquarerootofthearea. Noticethat(0.3)is aperfectanalogofKhinchin’slawoftheiteratedlogarithm in probability theory (describing the maximum fluctuations of the digit sums of a typical real number ˇ; the factor loglogn in (0.3) explains the name “iterated logarithm”). We also have an analogous central limit theorem: the renormalized counting function p F. 2IˇIcIen/(cid:3)2cn p ; 0(cid:2)ˇ <1; (cid:3) n hasastandardnormallimitdistributionwitherrortermO.n(cid:3)1=4.logn/3/asn!1 p [(cid:3) D(cid:3). 2;c/>0isthesamepositiveconstantasin(0.3)]. Preface ix Formally, ˇ n o ˇ p p maxˇmeasure ˇ 2Œ0;1/W F. 2IˇIcIen/(cid:3)2cn(cid:4)(cid:4)(cid:3) n (cid:4) Z ˇ (cid:3)p1 1e(cid:3)u2=2duˇˇˇDO(cid:6)(cid:2)n(cid:3)1=4.logn/3(cid:3); (0.5) 2(cid:5) (cid:4) wherethemaximumistakenoverall(cid:3)1<(cid:4)<1(andofcoursemeasuremeans theone-dimensionalLebesguemeasure). The proofs of the innocent-looking results (0.3) and (0.5) are quite difficult (in spite of the fact that most of the arguments are “elementary”). Note that here “independence”comesfrom a goodapproximationby modified Rademacher functions. The book is basically “lattice point counting” in disguise. This explains the subtitle randomness in lattice point counting. The main results are proved by the samescheme:werepresentanaturallatticepointcountingfunctionintheform X CX CX C::: C negligible; 1 2 3 whereX ;X ;X ;:::areindependentrandomvariables.Thiswaywecandirectly 1 2 3 applysomeclassicalresultsofprobabilitytheory(suchasthecentrallimittheorem and the law of the iterated logarithm). We have the following questions: (a) how toconstructtheindependentrandomvariablesX ;X ;X ;:::,(b)howtocompute 1 2 3 theexpectation,andfinally(c)howtocomputethevariance.Thesearesurprisingly difficultquestions. Ofcourse(0.3)and(0.5)extendtoallquadraticirrationals.Theyalsoextendto some other special numbers for which we know the continued expansion(e.g., e, p e2, e). Someofthemainresultsaboutquadraticirrationals(e.g.,Theorems1.1and1.2) do not extend to almost every ˛. The reason is that the continued fraction digits (officiallycalledpartialquotients)ofatypicalrealnumber˛exhibitaveryirregular behavior(seeSect.6.10). Someotherresults,including(0.3)and(0.5),dohaveananalogforalmostevery p p ˛.Thereis,however,adifference:thenormingfactor nisreplacedby nlogn, andalsotheerrortermismuchweaker(seeSect.6.10). Thekindof“randomness”weproveinthebookrequiressomeknowledgeabout the continued fraction expansion of the real number ˛. This is why the best way to demonstrate this “randomness” is to study the class of quadratic irrationals. Unfortunately,weknowverylittleaboutthecontinuedfractionofalgebraicnumbers of degree (cid:4) 3. This explains why we cannot prove anything about (say) the p “randomnessof 3 2”;thisiswhywecanprovestrongresultsaboutthe“randomness ofe,”andcanprovenothingaboutthe“randomnessof(cid:5).”

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This book gives a comprehensive treatment of random phenomena and distribution results in diophantine approximation, with a particular emphasis on quadratic irrationals. It covers classical material on the subject as well as many new results developed by the author over the past decade. A range of i
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