Continued fractions and transcendental numbers

Description

Continued fractions and transcendental numbers Boris ADAMCZEWSKI, Yann BUGEAUD, and Les DAVISON 1. Introduction It is widely believed that the continued fraction expansion of every irrational algebraic number ? either is eventually periodic (and we know that this is the case if and only if ? is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine in [22] (see also [6,39,41] for surveys including a discussion on this subject). A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The first result of this type goes back to the pioneering work of Liouville [26], who constructed transcendental real numbers with a very fast growing sequence of partial quo- tients. Indeed, the so-called ‘Liouville inequality' implies the transcendence of real numbers with very large partial quotients. Replacing it by Roth's theorem yields refined results, as shown by Davenport and Roth [15]. In [4], the argument of Davenport and Roth is slightly improved and Roth's theorem is replaced by a more recent result of Evertse [19] to obtain the best known result of this type. Note that the constant e, whose continued fraction expansion is given by e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, .

• folded continued


• numbers ?


• partial quo- tients


• bounded partial


• numbers


• logm logm


• real algebraic


• positive integers

Sujets

• Schmidt
• Thomas-Robert Bugeaud
• Numbers
• Natural number

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 14
• Langue: English

Extrait

Continuedfractionsandtranscendentalnumbers BorisA DAMCZEWSKI ,YannB UGEAUD ,andLesD AVISON 1.Introduction Itiswidelybelievedthatthecontinuedfractionexpansionofeveryirrationalalgebraic numberαeitheriseventuallyperiodic(andweknowthatthisisthecaseifandonlyifαis aquadraticirrational),oritcontainsarbitrarilylargepartialquotients.Apparently,this questionwasfirstconsideredbyKhintchinein[22](seealso[6,39,41]forsurveysincludinga discussiononthissubject).Apreliminarysteptowardsitsresolutionconsistsinproviding explicitexamplesoftranscendentalcontinuedfractions. ThefirstresultofthistypegoesbacktothepioneeringworkofLiouville[26],who constructedtranscendentalrealnumberswithaveryfastgrowingsequenceofpartialquo- tients.Indeed,theso-called‘Liouvilleinequality’impliesthetranscendenceofrealnumbers withverylargepartialquotients.ReplacingitbyRoth’stheoremyieldsrefinedresults, asshownbyDavenportandRoth[15].In[4],theargumentofDavenportandRothis slightlyimprovedandRoth’stheoremisreplacedbyamorerecentresultofEvertse[19] toobtainthebestknownresultofthistype.Notethattheconstante,whosecontinued fractionexpansionisgivenby e=[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,itstranscendencedoesnotfollowfromthecriteria from[26,15,4]. Attheoppositeside,thereisaquestforfindingexplicitexamplesoftranscendental continuedfractionswithboundedpartialquotients.Thefirstexamplesofsuchcontinued fractionswerefoundbyMaillet[28](seealsoSection34ofPerron[30]).Theproofof Maillet’sresultsisbasedonageneralformoftheLiouvilleinequalitywhichlimitsthe approximationofalgebraicnumbersbyquadraticirrationals.Theyweresubsequently improveduponbyA.Baker[10,11],whousedtheRoththeoremfornumberfieldsobtained byLeVeque[24].Lateron,Davison[16]appliedaresultofW.M.Schmidt[36],sayingina muchstrongerformthanLiouville’sinequalitythatarealalgebraicnumbercannotbewell approximablebyquadraticnumbers,toshowthetranscendenceofsomespecificcontinued fractions(seeSection4).Withthesameauxiliarytool,M.Queffe´lec[32]establishedthe niceresultthattheThue–Morsecontinuedfractionistranscendental(seeSection5).This methodhasthenbeenmademoreexplicit,andcombinatorialtranscendencecriteriabased onDavison’sapproachweregivenin[7,17,13,25]. 1