Boris ADAMCZEWSKI Lyon Yann BUGEAUD Strasbourg
16 Pages
English
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Boris ADAMCZEWSKI Lyon Yann BUGEAUD Strasbourg

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16 Pages
English

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Palindromic continued fractions Boris ADAMCZEWSKI (Lyon) & Yann BUGEAUD * (Strasbourg) 1. Introduction An old problem adressed by Khintchin [15] deals with the behaviour of the continued fraction expansion of algebraic real numbers of degree at least three. In particular, it is asked whether such numbers have or not arbitrarily large partial quotients in their con- tinued fraction expansion. Although almost nothing has been proved yet in this direction, some more general speculations are due to Lang [16], including the fact that algebraic numbers of degree at least three should behave like most of the numbers with respect to the Gauss–Khintchin–Kuzmin–Levy laws. A preliminary step consists in providing explicit examples of transcendental continued fractions. The first result of this type is due to Li- ouville [17], who constructed real numbers whose sequence of partial quotients grows very fast, too fast for being algebraic. Subsequently, various authors used deeper transcendence criteria from Diophantine approximation to construct other classes of transcendental con- tinued fractions. Of particular interest is the work of Maillet [18] (see also Section 34 of Perron [19]), who was the first to give examples of transcendental continued fractions with bounded partial quotients. Further examples were provided by A. Baker [8, 9], Davison [11], Queffelec [20], Allouche et al. [7], Adamczewski and Bugeaud [1, 5], and Adamczewski et al.

  • patterns being

  • arbitrarily large

  • rational numbers

  • palindrome

  • standard sturmian continued

  • partial quotients

  • numbers ?

  • results obtained


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PalindromiccontinuedfractionsBorisADAMCZEWSKI(Lyon)&YannBUGEAUD*(Strasbourg)1.IntroductionAnoldproblemadressedbyKhintchin[15]dealswiththebehaviourofthecontinuedfractionexpansionofalgebraicrealnumbersofdegreeatleastthree.Inparticular,itisaskedwhethersuchnumbershaveornotarbitrarilylargepartialquotientsintheircon-tinuedfractionexpansion.Althoughalmostnothinghasbeenprovedyetinthisdirection,somemoregeneralspeculationsareduetoLang[16],includingthefactthatalgebraicnumbersofdegreeatleastthreeshouldbehavelikemostofthenumberswithrespecttotheGauss–Khintchin–Kuzmin–Le´vylaws.Apreliminarystepconsistsinprovidingexplicitexamplesoftranscendentalcontinuedfractions.ThefirstresultofthistypeisduetoLi-ouville[17],whoconstructedrealnumberswhosesequenceofpartialquotientsgrowsveryfast,toofastforbeingalgebraic.Subsequently,variousauthorsuseddeepertranscendencecriteriafromDiophantineapproximationtoconstructotherclassesoftranscendentalcon-tinuedfractions.OfparticularinterestistheworkofMaillet[18](seealsoSection34ofPerron[19]),whowasthefirsttogiveexamplesoftranscendentalcontinuedfractionswithboundedpartialquotients.FurtherexampleswereprovidedbyA.Baker[8,9],Davison[11],Queffe´lec[20],Alloucheetal.[7],AdamczewskiandBugeaud[1,5],andAdamczewskietal.[6],amongothers.Acommonfeatureofalltheseresultsisthattheyapplytorealnumberswhosecontinuedfractionexpansionis‘quasi-periodic’inthesensethatitcontainsarbitrarilylongblocksofpartialquotientswhichoccurprecociouslyatleasttwice.Continuedfractionsbeginningwitharbitrarilylargepalindromesappearinseveralrecentpapers[21,22,10,13,2].Motivatedbythisandthegeneralproblematicmentionedabove,weaskwhetherprecociousoccurrencesofsomesymmetricpatternsinthecontinuedfractionexpansionofanirrationalrealnumberdoimplythatthelatteriseitherquadratic,ortranscendental.Weobtainthreenewtransendencecriteriathatapplytoabroadclassofcontinuedfractionexpansions,includingexpansionswithunboundedpartialquotients.Theseresultsprovidetheexactcounterpartof[1](seealsoTheorem3.1from[6]),withperiodicpatternsbeingreplacedbysymmetricones.Likein[1],theirproofsheavilydependontheSchmidtSubspaceTheorem[24].Asalreadymentioned,thereisalongtraditioninusinganexcessofperiodicitytoprovethetranscendenceofsomecontinuedfractions.Thisisindeedverynatural:ifthecontinuedfractionexpansionoftherealnumberξbeginswith,say,theperiodicpatternABBB(here,A,Bdenotetwofiniteblocksofpartialquotients),thenξis‘veryclose’tothequadraticirrationalrealnumberhavingtheeventuallyperiodic*SupportedbytheAustrianScienceFundationFWF,grantM822-N12.1