Bull London Math Soc C London Mathematical Society doi:10 blms bdq019

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Bull. London Math. Soc. 42 (2010) 538–552 C!2010 London Mathematical Society doi:10.1112/blms/bdq019 Rational numbers with purely periodic ?-expansion Boris Adamczewski, Christiane Frougny, Anne Siegel and Wolfgang Steiner Abstract We study real numbers ? with the curious property that the ?-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previously obtained in the case of degree 2 by Schmidt, Hama and Imahashi. Let ?(?) denote the supremum of the real numbers c in (0, 1) such that all positive rational numbers less than c have a purely periodic ?-expansion. We prove that ?(?) is irrational for a class of cubic Pisot units that contains the smallest Pisot number ?. This result is motivated by the observation of Akiyama and Scheicher that ?(?) = 0.666 666 666 086 . . . is surprisingly close to 2/3. 1. Introduction One of the most basic results about decimal expansions is that every rational number has an eventually periodic expansion (A sequence (an)n!1 is eventually periodic if there exists a positive integer p such that an+p = an for every positive integer n large enough), the converse being obviously true.

  • most basic results

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  • let ?

  • expansion according

  • positive real

  • vanishing ?-fractional

  • positive integer


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Bull.LondonMath.Soc.42(2010)538–552C!2010LondonMathematicalSocietydoi:10.1112/blms/bdq019Rationalnumberswithpurelyperiodic!-expansionBorisAdamczewski,ChristianeFrougny,AnneSiegelandWolfgangSteinerAbstractWestudyrealnumbers!withthecuriouspropertythatthe!-expansionofallsu!cientlysmallpositiverationalnumbersispurelyperiodic.ItisknownthatsuchrealnumbershavetobePisotnumberswhichareunitsofthenumberfieldtheygenerate.WecompleteknownresultsduetoAkiyamatocharacterizealgebraicnumbersofdegree3thatenjoythisproperty.Thisextendsresultspreviouslyobtainedinthecaseofdegree2bySchmidt,HamaandImahashi.Let"(!)denotethesupremumoftherealnumberscin(0,1)suchthatallpositiverationalnumberslessthanchaveapurelyperiodic!-expansion.Weprovethat"(!)isirrationalforaclassofcubicPisotunitsthatcontainsthesmallestPisotnumber#.ThisresultismotivatedbytheobservationofAkiyamaandScheicherthat"(#)=0.666666666086...issurprisinglycloseto2/3.1.IntroductionOneofthemostbasicresultsaboutdecimalexpansionsisthateveryrationalnumberhasaneventuallyperiodicexpansion(Asequence(an)n!1iseventuallyperiodicifthereexistsapositiveintegerpsuchthatan+p=anforeverypositiveintegernlargeenough),theconversebeingobviouslytrue.Infact,muchmoreisknownforwecaneasilydistinguishrationalswithapurelyperiodicexpansion(Asequence(an)n!1ispurelyperiodicifthereexistsapositiveintegerpsuchthatan+p=anforeverypositiveintegern):arationalnumberp/qintheinterval(0,1),inlowestform,hasapurelyperiodicdecimalexpansionifandonlyifqand10arerelativelyprime.Thus,bothrationalswithapurelyperiodicexpansionandrationalswithanon-purelyperiodicexpansionare,insomesense,uniformlyspreadontheunitinterval.Theseresultsextendmutatismutandistoanyintegerbaseb!2,asexplainedinthestandardmonographofHardyandWright[15].However,ifonereplacestheintegerbbyanalgebraicnumberthatisnotarationalinteger,itmayhappenthatthesituationwouldbedrasticallydi!erent.Asanillustrationofthisclaim,letusconsiderthefollowingtwoexamples.First,let!denotethegoldenratio,thatis,thepositiverootofthepolynomialx2!x!1.Everyrealnumber"in(0,1)canbeuniquelyexpandedasan=,"!n!1!nwhereantakesonlythevalues0and1,andwiththeadditionalconditionthatanan+1=0foreverypositiveintegern.Thebinarysequence(an)n!1istermedthe!-expansionof".In1980,Schmidt[22]provedtheintriguingresultthateveryrationalnumberin(0,1)hasapurelyperiodic!-expansion.Sucharegularityissomewhatsurprisingasonemayimagine!-expansionsofrationalsmoreintricatethantheirdecimalexpansions.Furthermore,thelatterpropertyseemstobequiteexceptional.Letusnowconsider#=1+!,thelargestrootofthepolynomialx2!3x+1.Again,everyrealnumber"in(0,1)hasa#-expansion,thatis,"canReceived7April2009;revised10November2009;publishedonline15April2010.2000MathematicsSubjectClassification11A63,11J72,11R06,28A80,37B50.DyTChiosNuwomr.khasbeensupportedbytheAgenceNationaledelaRecherche,grantANR-JCJC-06-134288