On the dichotomy of Perron numbers and beta conjugates

On the dichotomy of Perron numbers and beta conjugates

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On the dichotomy of Perron numbers and beta-conjugates Jean-Louis Verger-Gaugry? Prepublication de l'Institut Fourier no 709 (2008) Abstract Let ? > 1 be an algebraic number. A general definition of a beta-conjugate of ? is proposed with respect to the analytical function f?(z) = ?1 + ∑ i≥1 tiz i associated with the Renyi ?-expansion d?(1) = 0.t1t2 . . . of unity. From Szego's Theorem, we study the dichotomy problem for f?(z), in particular for ? a Perron number: whether it is a rational fraction or admits the unit circle as natural boundary. The first case of dichotomy meets Boyd's works. We introduce the study of the geometry of the beta-conjugates with respect to that of the Galois conjugates by means of the Erdo˝s-Turan approach and take examples of Pisot, Salem and Perron numbers which are Parry numbers to illustrate it. We discuss the possible existence of an infinite number of beta-conjugates and conjecture that all real algebraic numbers > 1, in particular Perron numbers, are in C1?C2?C3 after the classification of Blanchard/Bertrand-Mathis. Keywords: Perron number, Pisot number, Salem number, Erdo˝s-Turan's Theo- rem, numeration, Szego˝'s Theorem, uniform distribution, beta-shift, zeroes, beta- conjugate.

  • renyi ?-expansion

  • any galois

  • f?

  • perron numbers

  • erdo˝s-turan approach

  • let ?

  • theoreme de szego˝

  • following definition


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OnthedichotomyofPerronnumbersandbeta-conjugatesJean-LouisVerger-GaugryPre´publicationdel’InstitutFourierno709(2008)http://www-fourier.ujf-grenoble.fr/-Publications-.htmlAbstractLetβ>1beanalgebraicnumber.Ageneraldefinitionofabeta-cPonjugateiofβisproposedwithrespecttotheanalyticalfunctionfβ(z)=1+i1tizassociatedwiththeRe´nyiβ-expansiondβ(1)=0.t1t2...ofunity.FromSzego¨’sTheorem,westudythedichotomyproblemforfβ(z),inparticularforβaPerronnumber:whetheritisarationalfractionoradmitstheunitcircleasnaturalboundary.ThefirstcaseofdichotomymeetsBoyd’sworks.Weintroducethestudyofthegeometryofthebeta-conjugateswithrespecttothatoftheGaloisconjugatesbymeansoftheErdo˝s-Tura´napproachandtakeexamplesofPisot,SalemandPerronnumberswhichareParrynumberstoillustrateit.Wediscussthepossibleexistenceofaninfinitenumberofbeta-conjugatesandconjecturethatallrealalgebraicnumbers>1,inparticularPerronnumbers,areinC1C2C3aftertheclassificationofBlanchard/Bertrand-Mathis.Keywords:Perronnumber,Pisotnumber,Salemnumber,Erdo˝s-Tura´n’sTheo-rem,numeration,Szego˝’sTheorem,uniformdistribution,beta-shift,zeroes,beta-conjugate.Re´sume´Soitβ>1unnombrealge´brique.Onproposeunede´finitionge´ne´raled’unbeta-iPconjugue´deβenutilisantlafonctionanalytiquefβ(z)=1+i1tizassocie´eaude´veloppementdeRe´nyidβ(1)=0.t1t2...del’unite´.Apartirduthe´ore`medeSzego¨nouse´tudionsleproble`mededichotomiepourfβ(z),enparticulierpourβunnombredePerron:cettefonctionestsoitunefractionrationnelleouadmetlecercleunite´commebordnaturel.LepremiercasdedichotomierejointlestravauxdeBoyd.Nousintroduisonsl’e´tudecomparativedelage´ometriedesbeta-conjugue´setdecelledesconjugu´esdeGaloisaumoyendel’approched’Erdo˝s-Tura´netnousillustronscelle-cipardesexemplesdenombresdePisot,deSalemetdePerronquisontdesnombresdeParry.Nousdiscutonsl’existencepossibled’unnombreinfinidebeta-conjugue´sandconjecturonsquetouslesnombresalge´briquesre´els>1,enparticulierlesnombresdePerron,sontdansC1C2C3danslaclassificationdeBlanchard/Bertrand-Mathis.Mots-cle´s:nombredePerron,nombredePisot,nombredeSalem,The´ore`med’Erdo˝s-Tura´n,nume´ration,The´ore`medeSzego˝,e´quire´partition,beta-shift,ze´ros,beta-conjugue´.2000MathematicsSubjectClassification:11M99,03D45,30B10,12Y05.WorksupportedbyACINIM2004–154“Numeration”
2Pre´publicationdel’InstitutFourierno709–Janvier2008Contents1Introduction22Locusofzeroes53Beta-conjugates,Galoisconjugatesanddichotomy83.1Therationalfractioncase..........................83.2Thesecondcase:Hadamardtypetheorems,frequenciesofdigits....104Erdo˝s-Tura´napproach124.1Uniformclusteringneartheunitcircle...................124.2Examples...................................134.2.1SelmersPerronnumbersintheclassC1..............134.2.2BassinosinnitefamilyofcubicPisotnumbers..........154.2.3BoydsinnitefamiliesofPisotandSalemnumbers.......154.2.4ConuentParrynumbers......................171IntroductionThisnoteisconcernedwiththerichinterplaybetweennumbertheoryanddynamicalsystemsofnumeration,forwhichnumerationmeansnumerationinbaseβ,whereβisarealalgebraicnumber>1[2][7][8][31][38].Letusrecallsomedefinitionstofixnotations.Therealnumberβ>1isaPerronnumberifandonlyifitisanalgebraicintegerandallitsGaloisconjugatesβ(i)satisfy:|β(i)|foralli=1,2,...,d1,ifβisofdegreed1(withβ(0)=β).LetPbethesetofPerronnumbers.Wewillassumethroughoutthepaperthatthenumerationbasisβisnoninteger.TheRe´nyiβ-expansionof1isbydefinitiondenotedby+Xdβ(1)=0.t1t2t3...andcorrespondsto1=tiβi,(1.1)1=iwheret1=bβc,t2=bβ{β}c,t3=bβ{β{β}}c,...(bxcand{x}denotetheintegerpart,resp.thefractionalpart,ofarealnumberx).ThedigitstibelongtothefinitealphabetAβ={0,1,2,...,bβc}byconstruction.WewillsaythatβisaParrynumber(previouslycalledabeta-numberbyParry[36])ifdβ(1)isfiniteorultimatelyperiodic(i.e.eventuallyperiodic);inparticular,wesaythataParrynumberβissimpleifdβ(1)isfinite.Parry[36]hasshownthatParrynumbersarealgebraicintegers.Parrynumbers(simpleornon-simple)arePerronnumbers(Theorem7.2.13andProposition7.2.21inLothaire[31]).Theconverseiswrong,somehowmysterious.ThisdefinesadichotomyinP:thesubsetofPerronnumberswhichareParrynumbers,thesubsetofPwhicharenotParrynumbers.Themainmotivationofthepresentworkisthefollowing:ononehand,dβ(1)=0.t1t2...entirelycontrolstheβ-shift[11]makingthesequence(ti)i1averyimportantonewithvaluesinthealphabetAβ,ontheotherhandinthephilosophyofPo´lyaand