$Continued fractions and transcendental numbers$

Continued fractions and transcendental numbers

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folded continued

numbers ?

partial quo- tients

bounded partial

numbers

logm logm

real algebraic

positive integers

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profil-urra-2012

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ContinuedfractionsandtranscendentalnumbersBorisADAMCZEWSKI,YannBUGEAUD,andLesDAVISON1.IntroductionItiswidelybelievedthatthecontinuedfractionexpansionofeveryirrationalalgebraicnumberαeitheriseventuallyperiodic(andweknowthatthisisthecaseifandonlyifαisaquadraticirrational),oritcontainsarbitrarilylargepartialquotients.Apparently,thisquestionwasﬁrstconsideredbyKhintchinein[22](seealso[6,39,41]forsurveysincludingadiscussiononthissubject).Apreliminarysteptowardsitsresolutionconsistsinprovidingexplicitexamplesoftranscendentalcontinuedfractions.TheﬁrstresultofthistypegoesbacktothepioneeringworkofLiouville[26],whoconstructedtranscendentalrealnumberswithaveryfastgrowingsequenceofpartialquo-tients.Indeed,theso-called‘Liouvilleinequality’impliesthetranscendenceofrealnumberswithverylargepartialquotients.ReplacingitbyRoth’stheoremyieldsreﬁnedresults,asshownbyDavenportandRoth[15].In[4],theargumentofDavenportandRothisslightlyimprovedandRoth’stheoremisreplacedbyamorerecentresultofEvertse[19]toobtainthebestknownresultofthistype.Notethattheconstante,whosecontinuedfractionexpansionisgivenbye=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,...,1,1,2n,1,1,...],(seeforinstance[23])providesanexplicitexampleofatranscendentalnumberwithun-boundedpartialquotients;however,itstranscendencedoesnotfollowfromthecriteriafrom[26,15,4].Attheoppositeside,thereisaquestforﬁndingexplicitexamplesoftranscendentalcontinuedfractionswithboundedpartialquotients.TheﬁrstexamplesofsuchcontinuedfractionswerefoundbyMaillet[28](seealsoSection34ofPerron[30]).TheproofofMaillet’sresultsisbasedonageneralformoftheLiouvilleinequalitywhichlimitstheapproximationofalgebraicnumbersbyquadraticirrationals.TheyweresubsequentlyimproveduponbyA.Baker[10,11],whousedtheRoththeoremfornumberﬁeldsobtainedbyLeVeque[24].Lateron,Davison[16]appliedaresultofW.M.Schmidt[36],sayinginamuchstrongerformthanLiouville’sinequalitythatarealalgebraicnumbercannotbewellapproximablebyquadraticnumbers,toshowthetranscendenceofsomespeciﬁccontinuedfractions(seeSection4).Withthesameauxiliarytool,M.Queﬀe´lec[32]establishedtheniceresultthattheThue–Morsecontinuedfractionistranscendental(seeSection5).Thismethodhasthenbeenmademoreexplicit,andcombinatorialtranscendencecriteriabasedonDavison’sapproachweregivenin[7,17,13,25].1

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