A zero-one law for improvements to Dirichlet’s Theorem Dmitry Kleinbock.pdf

该【A zero-one law for improvements to Dirichlet’s Theorem Dmitry Kleinbock 】是由【周瑞】上传分享，文档一共【12】页，该文档可以免费在线阅读，需要了解更多关于【A zero-one law for improvements to Dirichlet’s Theorem Dmitry Kleinbock 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..PROCEEDINGSOFTHEAMERICANMATHEMATICALSOCIETYVolume146,Number5,May2018,Pages1833–1844http://dx./,2018AZERO-ONELAWFORIMPROVEMENTSTODIRICHLET’STHEOREMDMITRYKLEINBOCKANDNICKWADLEIGH(CommunicatedbyNimishShah)(oralmostno)x∈R,thesystem|qx?p|<ψ(t),|q|<tissolvableinp∈Z,q∈Z{0}forsu?,wecharacterizesuchxintermsofthegrowthoftheircontinuedfractionentries,andweestablishthatDirichlet’-,aswellasfornumerousendeavorsinthetheoryofDiophantineapproximation,isthefollowingtheorem,establishedbyDirichletin1842:(Dirichlet’sTheorem).Foranyx∈Randt>1,thereexistq∈Z{0},p∈Zsuchthat1()|qx?p|≤and|q|<,.,[Ca1,]or[Sc,].Inmanycasestheabovetheoremhasbeenappliedthroughitscorollary[Sc,],predatingDirichlet’swork:(Dirichlet’sCorollary).Foranyx∈Rthereexistin?nitelymanyq∈Zsuchthat1()|qx?p|<forsomep∈Z.|q|ThetwostatementsabovegivearateofapproximationwhichworksforallxandserveasabeginningofthemetrictheoryofDiophantineapproximation,()and()replacedbyfasterdecayingfunctionsoftand|q|,forafunctionψ:[t0,∞)→R+,wheret0≥1is?xed,letusde?neW(ψ),thesetofψ-approximablerealnumbers,tobethesetofx∈Rforwhichthereexistin?nitelymanyq∈Zsuchthat()|qx?p|<ψ(|q|)forsomep∈,2016,and,inrevisedform,January24,?,11J70;Secondary11J13,?rst-namedauthorwassupportedbyNSFgrantsDMS-1101320andDMS-1600814.:16:;seernal-terms-of-use:..1834DMITRYKLEINBOCKANDNICKWADLEIGHInwhatfollowswewillusethenotationψ1(t)=1/(ψ1)=>0suchthatW(cψ1)=,numberswhichdonotbelongtoW(cψ1)forsomec>0(equivalently,irrationalnumberswhosecontinuedfractioncoe?cientsareuniformlybounded)?dimension[J].HowevertheLebesguemeasureofthesetofbadlyapprox-imablenumbersiszero;inotherwords,W(cψ1)isco-nullforanyc>(ψ)(Khintchine’sTheorem).Givenanon-increasingψ,thesetW(ψ)haszero()measureifandonlyiftheserieskψ(k)converges().Quitesurprisingly,itseemsthatnosuchcleanstatementhasyetbeenprovedintheset-?nition:forψasabove,letD(ψ)denotethesetofx∈Rforwhichthesystem()|qx?p|<ψ(t)and|q|<thasanon-(ψ)willbecalledψ-?nitionarisesbyreplacing“≤ψ1(t)”in()with“<ψ(t)”anddemandingtheexistenceofnon-:?Ifψisnon-increasing,whichwillbeourstandingassumption,onecanwithoutlossofgeneralityrestricttot∈N:indeed,tosolve()itisenoughto?ndasolutionwithtreplacedbyt .?ItisnothardtoseethatD(ψ1)=R;moreprecisely,ifx/∈Q(∈Q),thesystem()withψ=ψ1hasanon-zerosolutionforallt>1(?cientlylarget).?ClearlyD(ψ)iscontainedinW(ψ)wheneverψisnon-,oneknowsthatD(ψ)andW(ψ)di?ersigni?cantlyforfunc-,ithasbeenobservedbyDavenportandSchmidt[DS1]thatthesetD(cψ1)ofcψ1-DirichletnumbershasLebesguemea-surezeroforanyc<,theyshowed[DS1,Theorem1]thatanirrationalnumberbelongstoD(cψ1)forsomec<∈c<1D(cψ1)ifandonlyifthecontinuedfractioncoe?:∈D(ψ)intermsofitscontinuedfractionexpansion?’sTheoremsharpinthesensethatifψ(t)<ψ1(t)forallsu?cientlylarget,thenthereexistsx∈Rwhichisnotψ-Dirichlet??cientconditiononψ(presumably,expressedintheformofconvergence/divergenceofacertainseries)guaranteeingthatthesetD(ψ)haszero/fullmeasure??→tψ(t):16:;seernal-terms-of-use:..AZERO-ONELAWFORIMPROVEMENTSTODIRICHLET’STHEOREM1835isnon-?cally,wewillprovethefollowing::[t0,∞)→R+isnon-increasingandψ(t)<ψ1(t)forsu?-cientlylarget,thenD(ψ)=:[t0,∞)→R+benon-increasing,andsupposethefunctiont→tψ(t)isnon-decreasingand()tψ(t)<1forallt≥?log1?nψ(n)1?nψ(n)()=∞(resp.<∞),nnthentheLebesguemeasureofD(ψ)((ψ)c)()isanaturalassumption:ifitisnotsatis?ed,thenD(ψ)=D(ψ1)=,takingψ=()equalto?clog(1?c).nnThuswerecovertheaforementionedresultofDavenportandSchmidtstatingthatD(cψ1)hasmeasurezeroforc<:11?at?k?Ifψ(t)=tfora>0,k≥0,thenthesumin()converges/divergesifandonlyifsodoes∞?k?k∞?log(t)tklogtdt=k+(ψ)hasfullmeasurewheneverk>?a(logt)?k?Ifψ(t)=tfora>0,k≥0,weareledtoconsider∞∞kloglogtklogukdt=(logt)1uInthiscaseD(ψ)hasfullmeasureifk>,followingsomelemmasexpressingtheψ-§3wediscussdynamicsoftheGaussmapx→1?1intheunitxxintervaland,following[Ph],establishadynamicalBorel----increasing,buteventuallydecreasing;clearlyonlytheeven-:16:;seernal-terms-of-use:..-quel,an=an(x)(n=1,2,...)willdenotethenthentryinthecontinuedfractionexpansionofx∈[0,1),andqn=qn(x),1pn[a1(x),a2(x),...,an(x)]=1=a1+a+1qn2...+a1nwithpn,qn∈=1,{qn}∞maybede?nedasthen=0increasingsequenceofpositiveintegerswiththepropertyqnx<qxforallpositiveintegersq<{an},{qn}arerelatedbytherecurrence()qn=anqn?1+qn?[Kh]orthe?rstchapterof[Ca](x),an(x)donotterminate;thatis,weexcludethecasex∈,wewillonlyconsiderx∈[0,1):[t0,∞)→R+benon-∈[0,1]Qisψ-Dirichletifandonlyifqn?1x<ψ(qn)forsu?∈[0,1]Qisψ-?cientlylargenthereexistsapositiveinteger,q,withqx<ψ(qn),q<?1x≤qxwheneverq<qn,wehaveqn?1x<ψ(qn)forsu?,supposeqn?1x<ψ(qn)forn≥>qN,writeqn?1<t≤?1x<ψ(t)followssinceψisnon--Dirichlet.-Dirichletpropertyofxintermsofthegrowthofthecontinuedfractionentries,an(x).For?xedx=[a1,a2,...],considerthesequencesθn+1=[an+1,an+2,...],φn=[an,an?1,...,a1].Thesearerelatedtothesequencesqn,qn?1xbytheidentity()(1+θn+1φn)?1=qnqn?1x(see[Ca1,§]).,∈[0,1]Q,andletψ:[t0,∞)→R+benon-increasingwithtψ(t)<1forallt≥?1 ?1(i)xisψ-Dirichletifan+1an≤qnψ(qn)?1forallsu?’de?nitionofthesequence{qn}di?,Casselshasq1=1,q2=a1,....WehaveadoptedKhintchine’,Cassels’formulareads(1+θn+1φn)?1=qn+1qnxbecausehisqn’sareshiftedbyoneindex,:16:;seernal-terms-of-use:..AZERO-ONELAWFORIMPROVEMENTSTODIRICHLET’STHEOREM1837?1 ?1(ii)xisnotψ-Dirichletifan+1an>qnψ(qn)?1forin?∈[0,1](),()x∈D(ψ)ifandonlyif(1+θn+1φn)?1<qnψ(qn)(an+1+1)(an+1)≤4an+1an,wehavean+2an?1?111?11+·<(1+θn+1φn)an+1an?1?1111<1+1·1≤1+.an+1+an+2an+an?14an+1anHencefrom(),x∈D(ψ)if?111+≤qnψ(qn)4an+1anforsu??rstassertionofthelemmabysolvingforanan+,x/∈D(ψ)if?111+>qnψ(qn)an++1givesthesecondassertionofthelemma.[DS1,Theorem1]statingthatx∈D(cψ1)forsomec<1ifandonlyifthesequencean(x)-Dirichletforanynon-increasingψwithψ(t)<ψ1(t)forsu?(),qndependsonlyona1,...,(t)<1forlargeenought,wemayconstructx=[a1,a2,...]essivelychoosingan+1sothatpart(ii)?ed.,,()isinsolublewhent=qnforallsu?cientlylargen–notjustforin?-CantelliLemmasForalmosteveryx,wehavereducedtheψ-,()T:[0,1]Q→[0,1]Q,x→x?1?x?1,hastheconvenientpropertyT([a1,a2,a3,...])=[a2,a3,a4....],anditpreservestheGaussmeasure,11()μ(A)=+:16:;seernal-terms-of-use:..1838DMITRYKLEINBOCKANDNICKWADLEIGHWewillusetworesultsofPhilipp[Ph]relatedtothemixingrateofTandthedivergencecaseoftheBorel-([Ph,]).LetEn,n≥1,beasequenceofmeasurablesetsinaprobabilityspace(X,ν).DenotebyA(N,x)thenumberofintegersn≤Nsuchthatx∈φ(N)=ν(En).n≤NSupposethatthereexistsaconvergentseriesj≥1CjwithCj≥0suchthatforallintegersm>nwehave()ν(En∩Em)≤ν(En)ν(Em)+ν(Em)Cm?>0onehas1/23/2+A(N,x)=φ(N)+Oφ(N)logφ(N)|ν(En∩Em)?ν(En)ν(Em)|≤ν(Em),-iallystrengthened:givenany>0,theconclusionofthetheoremholdsprovidedtheinequality()holdswheneverm>n+.([Ph,]).Thereexistconstantsc0>0and0<γ<=(r1,...,rk)∈Nk,andwriteEr:={x∈[0,1]Q:a1(x)=r1,a2(x)=r2,...,ak(x)=rk}.LetF?[0,1]≥0,?n?k√n()μ(Er∩TF)?μ(Er)μ(F)≤c0μ(Er)μ(F),thisestimateadmitspassingtounions:?[0,1]∈N,andletR?()holdsforalln≥0whenErisreplacedwith∪r∈?n?kμEr∩TF?μErμ(F)r∈Rr∈R?n?k=μ(Er∩TF)?μ(Er)μ(F)r∈R√n√n≤c0μ(Er)μ(F)γ=c0μErμ(F)∈Rr∈RheabovestatementstoestablishaquitegeneraldynamicalBorel-CantelliLemma:∈(n∈N)isasequenceofsetssuchthateachAnisaunionofsetsoftheformEr,r∈Nk(Erasde?).Ifnμ(An)=∞(resp.<∞),thenforalmostevery()x∈[0,1]onehasTn(x)∈Anforin?nitelymanyn.