On the dichotomy of Perron numbers and beta conjugates

Jean-Louis Verger-Gaugry

Description

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

Informations

• Publié par: pefav
• Nombre de lectures: 8
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

OnthedichotomyofPerronnumbersandbeta-conjugates Jean-LouisVerger-Gaugry ∗ Pre´publicationdel’InstitutFouriern o 709(2008) http://www-fourier.ujf-grenoble.fr/-Publications-.html Abstract Letβ>1beanalgebraicnumber.Ageneraldefinitionofabeta-c P onjugate i ofβisproposedwithrespecttotheanalyticalfunctionf β (z)= − 1+ i ≥ 1 t i z associatedwiththeRe´nyiβ-expansiond β (1)=0.t 1 t 2 ...ofunity.FromSzego¨’s Theorem,westudythedichotomyproblemforf β (z),inparticularforβaPerron number:whetheritisarationalfractionoradmitstheunitcircleasnatural boundary.ThefirstcaseofdichotomymeetsBoyd’sworks.Weintroducethe studyofthegeometryofthebeta-conjugateswithrespecttothatoftheGalois conjugatesbymeansoftheErdo˝s-Tura´napproachandtakeexamplesofPisot, SalemandPerronnumberswhichareParrynumberstoillustrateit.Wediscuss thepossibleexistenceofaninfinitenumberofbeta-conjugatesandconjecturethat allrealalgebraicnumbers>1,inparticularPerronnumbers,areinC 1 ∪ C 2 ∪ C 3 aftertheclassificationofBlanchard/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- i P conjugue´deβenutilisantlafonctionanalytiquef β (z)= − 1+ i≥1 t i zassocie´e aude´veloppementdeRe´nyid β (1)=0.t 1 t 2 ...del’unite´.Apartirduthe´ore`mede Szego¨nouse´tudionsleproble`mededichotomiepourf β (z),enparticulierpourβ unnombredePerron:cettefonctionestsoitunefractionrationnelleouadmetle cercleunite´commebordnaturel.Lepremiercasdedichotomierejointlestravaux deBoyd.Nousintroduisonsl’e´tudecomparativedelage´ometriedesbeta-conjugue´s etdecelledesconjugu´esdeGaloisaumoyendel’approched’Erdo˝s-Tura´netnous illustronscelle-cipardesexemplesdenombresdePisot,deSalemetdePerronqui sontdesnombresdeParry.Nousdiscutonsl’existencepossibled’unnombreinfini debeta-conjugue´sandconjecturonsquetouslesnombresalge´briquesre´els>1,en particulierlesnombresdePerron,sontdansC 1 ∪ C 2 ∪ C 3 danslaclassificationde Blanchard/Bertrand-Mathis. Mots-cle´s:nombredePerron,nombredePisot,nombredeSalem,The´ore`me d’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”

2 Pre´publicationdel’InstitutFouriern o 709–Janvier2008 Contents 1Introduction2 2Locusofzeroes5 3Beta-conjugates,Galoisconjugatesanddichotomy8 3.1Therationalfractioncase..........................8 3.2Thesecondcase:Hadamardtypetheorems,frequenciesofdigits....10 4Erdo˝s-Tura´napproach12 4.1Uniformclusteringneartheunitcircle...................12 4.2Examples...................................13 4.2.1Selmer’sPerronnumbersintheclassC 1 ..............13 4.2.2Bassino’sinfinitefamilyofcubicPisotnumbers..........15 4.2.3Boyd’sinfinitefamiliesofPisotandSalemnumbers.......15 4.2.4ConfluentParrynumbers......................17 1Introduction Thisnoteisconcernedwiththerichinterplaybetweennumbertheoryanddynamical systemsofnumeration,forwhichnumerationmeansnumerationinbaseβ,whereβis arealalgebraicnumber>1[2][7][8][31][38]. Letusrecallsomedefinitionstofixnotations.Therealnumberβ>1isaPerron numberifandonlyifitisanalgebraicintegerandallitsGaloisconjugatesβ (i) satisfy: | β (i) | <βforalli=1,2,...,d − 1,ifβisofdegreed ≥ 1(withβ (0) =β).Let P be thesetofPerronnumbers.Wewillassumethroughoutthepaperthatthenumeration basisβisnoninteger.TheRe´nyiβ-expansionof1isbydefinitiondenotedby ∞ + X d β (1)=0.t 1 t 2 t 3 ...andcorrespondsto1=t i β − i ,(1.1) 1=i wheret 1 = b β c ,t 2 = b β { β }c ,t 3 = b β { β { β }}c ,...( b x c and { x } denotetheinteger part,resp.thefractionalpart,ofarealnumberx).Thedigitst i belongtothefinite alphabet A β = { 0,1,2,..., b β c} byconstruction.WewillsaythatβisaParrynumber (previouslycalledabeta-numberbyParry[36])ifd β (1)isfiniteorultimatelyperiodic (i.e.eventuallyperiodic);inparticular,wesaythataParrynumberβissimpleifd β (1) isfinite. Parry[36]hasshownthatParrynumbersarealgebraicintegers.Parrynumbers (simpleornon-simple)arePerronnumbers(Theorem7.2.13andProposition7.2.21in Lothaire[31]).Theconverseiswrong,somehowmysterious.Thisdefinesadichotomy in P :thesubsetofPerronnumberswhichareParrynumbers,thesubsetof P which arenotParrynumbers. Themainmotivationofthepresentworkisthefollowing:ononehand,d β (1)= 0.t 1 t 2 ...entirelycontrolstheβ-shift[11]makingthesequence(t i ) i ≥ 1 averyimportant onewithvaluesinthealphabet A β ,ontheotherhandinthephilosophyofPo´lyaand