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Uniform Distribution of Galois Conjugates and Beta-Conjugates of a Parry Number Near the Unit Circle and Dichotomy of Perron Numbers Jean-Louis Verger-Gaugry

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Uniform Distribution of Galois Conjugates and Beta-Conjugates of a Parry Number Near the Unit Circle and Dichotomy of Perron Numbers Jean-Louis Verger-Gaugry Uniform distribution of Galois conjugates and beta-conjugates of a Parry number near the unit circle and dichotomy of Perron numbers Jean-Louis Verger-Gaugry To cite this version: Jean-Louis Verger-Gaugry. Uniform distribution of Galois conjugates and beta-conjugates of a Parry number near the unit circle and dichotomy of Perron numbers. Uniform Distribution Theory, Math- ematical Institute of the Slovak Academy of Sciences, 2009, 3 (2), pp.157–190. hal-00341880 HAL Id: hal-00341880 https://hal.archives-ouvertes.fr/hal-00341880 Submitted on 26 Nov 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Uniform distribution of Galois conjugates and beta-conjugates of a Parry number near the unit circle and dichotomy of Perron numbers Jean-Louis Verger-Gaugry Institut Fourier, CNRS UMR 5582, Universit´ede Grenoble I, BP 74, 38402 Saint-Martin d’H`eres, France email: [email protected] Abstract. Concentration and equi-distribution, near the unit circle, in Solomyak’s set, of the union of the Galois conjugates and the beta-conjugates of a Parry number β are characterized by means of the Erd˝os-Tur´an approach, and its improvements by i Mignotte and Amoroso, applied to the analytical function fβ (z) = −1+ i≥1 tiz associated with the R´enyi β-expansion dβ (1) = 0.t1t2 . of unity. Mignotte’s dis- crepancy function requires the knowledge of the factorization of the ParryP polyno- mial of β. This one is investigated using theorems of Cassels, Dobrowolski, Pinner and Vaaler, Smyth, Schinzel in terms of cyclotomic, reciprocal non-cyclotomic and non-reciprocal factors. An upper bound of Mignotte’s discrepancy function which arises from the beta-conjugates of β which are roots of cyclotomic factors is linked to the Riemann hypothesis, following Amoroso. An equidistribution limit theorem, following Bilu’s theorem, is formulated for the concentration phenomenon of conju- gates of Parry numbers near the unit circle. Parry numbers are Perron numbers. Open problems on non-Parry Perron numbers are mentioned in the context of the existence of non-unique factorizations of elements of number fields into irreducible Perron numbers (Lind). 2000 Mathematics Subject Classification: 11M99, 30B10, 12Y05. Keywords: Parry number, Perron number, Pisot number, Salem number, numera- tion, beta-integer, beta-shift, zeroes, beta-conjugate, discrepancy, Erd˝os-Tur´an, Rie- mann hypothesis, Parry polynomial, factorization, equidistribution. 2 Jean-Louis Verger-Gaugry Contents 1 Introduction ............................... 2 2 Szeg˝o’sTheoremandnumeration . 4 3 Parry polynomials, Galois- and beta-conjugates in Solomyak’s set Ω for a Parry number β .......................... 5 3.1 Erd˝os-Tur´an’s approach and Mignotte’s discrepancy splitting . 5 3.2 Factorizationandirreduciblefactors . 12 3.3 Cyclotomicfactors .. .. .. .. .. .. .. .. .. .. .. 16 3.4 Non-cyclotomicfactors. 20 3.5 RealpositiveconjugatesofaParrynumber . 22 3.6 An equidistribution limit theorem . 24 4 Examples................................. 26 5 Arithmetics of Perronnumbers and non-Parrycase . 28 1 Introduction A Perron number is either 1 or a real number β > 1 which is an algebraic integer such that all its Galois conjugates β(i) satisfy: β(i) < β for all i = 1, 2,...,d 1, if β is of degree d 1 (with β(0) = β).| Let| P be the set of Perron numbers.− This set P is partitioned≥ into two disjoint components whose frontier is badly known [V2]. This partitioning arises from the properties of the numeration in base β, i.e. of the β-shifts and the dynamical systems ([0, 1],Tβ) [Bl] [Fr1], where β runs over (1, + ), and where Tβ(x) = βx is the beta- transformation ( x , x and x denote∞ the integer part, resp.{ } the fractional part, resp. the smallest⌊ ⌋ { } integer⌈ greater⌉ than or equal to a real number x). Let us recall this dichotomy and fix some notations. Let β > 1 be a real number and assume throughout the paper that β is non-integer. The R´enyi β-expansion of 1 is by definition denoted by + ∞ i dβ(1) = 0.t1t2t3 . and correspondsto 1= tiβ− , (1.1) i=1 X 2 where t1 = β ,t2 = β β = βTβ(1) ,t3 = β β β = βTβ (1) ,... The digits t belong⌊ ⌋ to the⌊ { finite}⌋ alphabet⌊ ⌋ := ⌊0{, 1,{2,...,}}⌋ β⌊ 1 . ⌋β is said i Aβ { ⌈ − ⌉} to be a Parry number if dβ(1) is finite or ultimately periodic (i.e. eventually periodic); in particular, a Parry number β is said to be simple if dβ(1) is finite. A proof that Parry numbers are Perron numbers is given by Theorem 7.2.13 and Proposition 7.2.21 in Lothaire [Lo]. On the contrary a good proportion of Perron numbers are not Parry numbers, so that the dichotomy of P can be Conjugates of a Perron number and Erd˝os-Tur´an approach 3 stated as P = P P , (1.2) P ∪ a where PP denotes the set of Parry numbers and Pa the set of Perron numbers which are not Parry numbers (a stands for aperiodic). The set PP is dense in (1, + ) [Pa], contains all Pisot numbers [B] [Bo2] [St] and Salem numbers of degree∞ 4 [Bo1], at least [Bo3] [V2]. Following [V2] the present note continues the exploration of this dichotomy by the Erd˝os-Tur´an approach applied to the collection := (fβ(z))β P of analytic functions B ∈ + ∞ f (z) := t zi for β P,z C, (1.3) β i ∈ ∈ i=0 X with t = 1, canonically associated with the R´enyi expansions d (1) = 0 − β 0.t1t2t3 . ., for which fβ(z) is a rational fraction if and only if β PP (Section 2); when β P the opposite of the reciprocal polynomial of the∈ numerator of ∈ P fβ(z), namely nβ∗ (X), is the Parry polynomial of β, a multiple of the minimal polynomial of β. Section 3 explores the factorization of nβ∗ (X), the geometry and the equi- distribution of its roots near the unit circle when β is a Parry number. These roots are either Galois conjugates or beta-conjugates of β and comparison is made between these two collections of roots. The important points in this exploration are: (i) the discrepancy function as obtained by Mignotte [G] [Mt1] [Mt2] in a theorem which generalizes Erd˝os-Tur´an’s Theorem, its splitting and its properties following Amoroso [A2] [AM], how the beta-conjugates of β which are roots of unity, as roots of the irreducible cyclotomic factors of nβ∗ (X), are linked to the Riemann hypothesis (R.H.), (ii) the number of positive real (Galois- or beta-) conjugates of β when the degree of β is large, by comparison with Kac’s formula [K] [EK], (iii) upper bounds for the multiplicities of beta- conjugates, (iv) an equidistribution limit theorem in the same formulation as Bilu’s Theorem [Bi] for convergent sequences of Parry numbers (or Parry polynomials) with Haar measure on the unit circle as limit measure. Mignotte’s discrepancy function allows a much better strategy than Erd˝os- Tur´an’s discrepancy function in the Erd˝os-Tur´an approach of the Parry poly- nomial nβ∗ (X) since it is a subaddtive function on its factorizaton and enables to investigate the roles played by its factors, namely cyclotomic, reciprocal non-cyclotomic and non-reciprocal, term by term. The factorization of the Parry polynomial nβ∗ (X) is itself a formidable chal- lenge because of the difficulty of determining the types of its factors, their multiplicities and the way it is correlated to the Rauzy fractal (central tile) (Barat, Berth´e, Liardet and Thuswaldner [B-T]). The problem of Lehmer of finding the smallest Mahler measure is essentially equivalent to the problem of 4 Jean-Louis Verger-Gaugry estimating the number of irreducible non-cyclotomic factors of nβ∗ (X) (Pinner and Vaaler [PV3]). Section 4 provides various examples on which Erd˝os-Tur´an’s and Mignotte’s discrepancy functions are computed. In Section 5 the case of Perron numbers which are not Parry numbers is evoked in the general context of the arithmetics of Perron numbers where non-unique factorizations on irreducible Perron num- bers may occur. d 1 (i) Notations: N(β)= i=0− β is the algebraic norm of the algebraic number (0) β(= β ), of degree d 1; Pβ(X) is the minimal polynomial of the algebraic Q≥ m number β > 1, with positive leading coefficient; R∗(X) = X R(1/X) is the reciprocal polynomial of the polynomial R(X) (of degree m) and R(X) is said 1/2 m 2 reciprocal if R(X)= R∗(X); Q 2 = j=0 αj , resp. Q 1 = L(Q)= m || || | | || || αj , H(Q) = max0 j m αj , the 2-norm, resp. the 1-norm (or length), j=0 | | ≤ ≤ | | P resp. the height, of the polynomial Q(X) = m α Xj, α C; M(R) = P j=0 j j ∈ a m max 1, b denotes the Mahler measure of the polynomial R(X)= | R| j=0 { | j|} P a m (X b ) C[X] where a is the leading coefficient. D(z , r) denotes R Qj=0 − j ∈ R 0 C + theQ open disk centred at z0 of radius r > 0, D(z0, r) its closure. Log x (x > 0) denotes max Log x,∈0 . The constants implied by the Vinogradov { } symbol ‘ ’ are absolute and computable; when written ‘ ǫ’ for ǫ > 0, they depend upon≪ ǫ.
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