Real and p adic expansions involving symmetric patterns

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Niveau: Supérieur, Doctorat, Bac+8
Real and p-adic expansions involving symmetric patterns Boris ADAMCZEWSKI & Yann BUGEAUD Abstract. This paper is motivated by the non-Archimedean counter- part of a problem raised by Mahler and Mendes France, and by questions related to the expected normality of irrational algebraic numbers. We introduce a class of sequence enjoying a particular combinatorial prop- erty: the precocious occurrences of infinitely many symmetric patterns. Then, we prove several transcendence statements involving both real and p-adic numbers associated with these palindromic sequences. 1. Introduction One motivation for the present paper comes from the following question concerning the expansion of algebraic numbers in integer bases. It appears at the end of a paper of Mendes France [8], but in conversation he attributes the paternity of this problem to Mahler (see the discussion in [4], page 403). Though we do not find any trace of it in Mahler's work, we will refer to it as the Mahler–Mendes France problem. It can be stated as follows: For an arbitrary infinite sequence a = (ak)k≥1 of 0's and 1's, prove that the real numbers +∞ ∑ k=1 ak 2k and +∞ ∑ k=1 ak 3k are algebraic if and only if both are rational. We raise here a non-Archimedean version of this conjecture.

  • sequence

  • main transcendence

  • numbers

  • monna map

  • infinitely many symmetric

  • find any

  • sequences

  • irrational algebraic

  • k≥?k0 ak


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Nombre de lectures 46
Langue English
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Realandp-adicexpansionsinvolvingsymmetricpatternsBorisADAMCZEWSKI&YannBUGEAUDAbstract.Thispaperismotivatedbythenon-Archimedeancounter-partofaproblemraisedbyMahlerandMende`sFrance,andbyquestionsrelatedtotheexpectednormalityofirrationalalgebraicnumbers.Weintroduceaclassofsequenceenjoyingaparticularcombinatorialprop-erty:theprecociousoccurrencesofinfinitelymanysymmetricpatterns.Then,weproveseveraltranscendencestatementsinvolvingbothrealandp-adicnumbersassociatedwiththesepalindromicsequences.1.IntroductionOnemotivationforthepresentpapercomesfromthefollowingquestionconcerningtheexpansionofalgebraicnumbersinintegerbases.ItappearsattheendofapaperofMende`sFrance[8],butinconversationheattributesthepaternityofthisproblemtoMahler(seethediscussionin[4],page403).ThoughwedonotfindanytraceofitinMahler’swork,wewillrefertoitastheMahler–Mende`sFranceproblem.Itcanbestatedasfollows:Foranarbitraryinfinitesequencea=(ak)k1of0’sand1’s,provethattherealnumbers+Xak+Xakdnakkk=12k=13arealgebraicifandonlyifbotharerational.Weraisehereanon-Archimedeanversionofthisconjecture.Problem1.Letpbeaprimenumber,a=(ak)k1beaninfinitesequenceon{0,1,...,p1},andsetk+Xa+Xkα=kandαp=akp.pk=1k=1Then,provethattherealnumberαandthep-adicnumberαparealgebraicifandonlyifbotharerational.AnotherformulationofProblem1isthatforeverynon-eventuallyperiodicsequenceaatleastonenumberamongαandαpistranscendental.Tothebestofourknowledge,noexampleisknownofsuchasequenceforwhichweareabletoreachthisconclusion2000MathematicsSubjectClassification:11J81,11J61.1