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Uniform Distribution Theory 3 (2008), No.2, 157–190 Distribution Theory uniform Uniform Distribution Theory 3 (2008), no.2, 157{190 distribution theory 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 ABSTRACT. Concentration and equi-distribution, near the unit circle, in Solo- myak's set, of the union of the Galois conjugates and the beta-conjugates of a Parry number ¯ are characterized by means of the Erd}os-Tur¶anapproach, and its improvements by Mignotte and Amoroso, applied to the analytical function P i f¯ (z) = ¡1 + i¸1 tiz associated with the R¶enyi ¯-expansion d¯ (1) = 0:t1t2 ::: of unity. Mignotte's discrepancy function requires the knowledge of the factor- ization of the Parry polynomial of ¯. This one is investigated using theorems of Cassels, Dobrowolski, Pinner and Vaaler, Smyth, Schinzel in terms of cyclo- tomic, reciprocal non-cyclotomic and non-reciprocal factors. An upper bound of Mignotte's discrepancy function which arises from the beta-conjugates of ¯ which are roots of cyclotomic factors is linked to the Riemann hypothesis, following Amoroso. An equidistribution limit theorem, following Bilu's theorem, is formu- lated for the concentration phenomenon of conjugates 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 fac- torizations of elements of number ¯elds into irreducible Perron numbers (Lind). Communicated by Ilya Shkredov Contents 1. Introduction 158 2. Szeg}o'sTheorem and numeration 160 3. Parry polynomials, Galois- and beta-conjugates in Solomyak's set ­ for a Parry number ¯ 161 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: 11M99, 30B10, 12Y05. K e y w o r d s: Parry number, Perron number, Pisot number, Salem number, numeration, beta- integer, beta-shift, zeroes, beta-conjugate, discrepancy, Erd}os-Tur¶an,Riemann hypothesis, Parry polynomial, factorization, equidistribution. 157 JEAN-LOUIS VERGER-GAUGRY 3.1. Erd}os-Tur¶an'sapproach and Mignotte's discrepancy splitting 161 3.2. Factorization and irreducible factors 168 3.3. Cyclotomic factors 173 3.4. Non-cyclotomic factors 178 3.5. Real positive conjugates of a Parry number 179 3.6. An equidistribution limit theorem 182 4. Examples 183 5. Arithmetics of Perron numbers and non-Parry case 185 REFERENCES 188 1. Introduction A Perron number is either 1 or a real number ¯ > 1 which is an algebraic integer such that all its Galois conjugates ¯(i) satisfy: j¯(i)j < ¯ for all i = 1; 2; : : : ; d ¡ 1, if ¯ is of degree d ¸ 1 (with ¯(0) = ¯). 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 ¯, i.e., of the ¯-shifts and the dynamical systems ([0; 1];T¯) [Bl] [Fr1], where ¯ runs over (1; +1), and where T¯(x) = f¯xg is the beta-transformation (bxc, fxg and dxe 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 ¯x some notations. Let ¯ > 1 be a real number and assume throughout the paper that ¯ is non-integer. The R¶enyi ¯-expansion of 1 is by de¯nition denoted by X+1 ¡i d¯(1) = 0:t1t2t3 ::: and corresponds to 1 = ti¯ ; (1.1) i=1 2 where t1 = b¯c; t2 = b¯f¯gc = b¯T¯(1)c; t3 = b¯f¯f¯ggc = b¯T¯ (1)c;::: The digits ti belong to the ¯nite alphabet A¯ := f0; 1; 2;:::; d¯ ¡ 1eg. ¯ is said to be a Parry number if d¯(1) is ¯nite or ultimately periodic (i.e., eventually periodic); in particular, a Parry number ¯ is said to be simple if d¯(1) is ¯nite. 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 stated as P = PP [ Pa; (1.2) 158 CONJUGATES OF A PERRON NUMBER AND ERDOS-TUR} AN¶ APPROACH 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; +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¶anapproach applied to the collection B := (f¯(z))¯2P of analytic functions X+1 i f¯(z) := tiz for ¯ 2 P; z 2 C; (1.3) i=0 with t0 = ¡1, canonically associated with the R¶enyi expansions d¯(1) = 0:t1t2t3 :::; for which f¯(z) is a rational fraction if and only if ¯ 2 PP (Sec- tion 2); when ¯ 2 PP the opposite of the reciprocal polynomial of the numerator ¤ of f¯(z), namely n¯(X), is the Parry polynomial of ¯, a multiple of the minimal polynomial of ¯. ¤ Section 3 explores the factorization of n¯(X), the geometry and the equi- distribution of its roots near the unit circle when ¯ is a Parry number. These roots are either Galois conjugates or beta-conjugates of ¯ and comparison is made between these two collections of roots. The important points in this ex- ploration are: (i) the discrepancy function as obtained by Mignotte [G] [Mt1] [Mt2] in a theorem which generalizes Erd}os-Tur¶an'sTheorem, its splitting and its properties following Amoroso [A2] [AM], how the beta-conjugates of ¯, which ¤ are roots of unity, as roots of the irreducible cyclotomic factors of n¯(X), are linked to the Riemann hypothesis (R.H.), (ii) the number of positive real (Galois- or beta-) conjugates of ¯ when the degree of ¯ 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 Theo- rem [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'sdiscrepancy function in the Erd}os-Tur¶anapproach of the Parry poly- ¤ nomial n¯(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 n¯(X) is itself a formidable chal- lenge because of the di±culty of determining the types of its factors, their mul- tiplicities 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 ¯nding 159 JEAN-LOUIS VERGER-GAUGRY the smallest Mahler measure is essentially equivalent to the problem of esti- ¤ mating the number of irreducible non-cyclotomic factors of n¯(X) (Pinner and Vaaler [PV3]). Section 4 provides various examples on which Erd}os-Tur¶an'sand 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 numbers may occur. Qd¡1 (i) Notations: N(¯) = i=0 ¯ is the algebraic norm of the algebraic number (0) ¯(= ¯ ), of degree d ¸ 1; P¯(X) is the minimal polynomial of the algebraic number ¯ > 1, with positive leading coe±cient; R¤(X) = XmR(1=X) is the reciprocal polynomial of the polynomial R(X) (of degree m) and R(X) is said ³ ´1=2 ¤ Pm 2 reciprocal if R(X) = R (X); jjQjj2 = j=0 j®jj , resp. jjQjj1 = L(Q) = Pm j=0 j®jj, H(Q) = max0·j·m j®jj, the 2-norm, resp. the 1-norm (or length), Pm j resp. the height, of the polynomial Q(X) = j=0 ®jX , ®j 2 C; M(R) = Qm jaRj j=0 maxf1; jbjjg denotes the Mahler measure of the polynomial R(X) = Qm aR j=0(X ¡bj) 2 C[X] where aR is the leading coe±cient. D(z0; r) denotes the + open disk centred at z0 2 C of radius r > 0, D(z0; r) its closure. Log x (x > 0) denotes maxfLog x; 0g. The constants implied by the Vinogradov symbol `¿' are absolute and computable; when written `¿²' for ² > 0, they depend upon ². 2. Szeg}o'sTheorem and numeration Every analytical function f¯(z) with ¯ > 1 (¯ 62 N) obeys the dichotomy given by the following theorem ([Sg], [Di] p 324{7) since its coe±cients belong to A¯, which is ¯nite. P Theorem . n 2.1 (Szeg}o) A Taylor series n¸0 anz with coe±cients in a ¯nite subset S of C is either equal to V (z) P (i) a rational fraction U(z) + zm+1 where U(z) = ¡1 + m b zi , 1 ¡ zp+1 i=1 i Pp i V (z) = i=0 eiz are polynomials with coe±cients in S and m ¸ 1; p ¸ 0 integers, or (ii) it is an analytic function de¯ned on the open unit disk which is not con- tinued beyond the unit circle (which is its natural boundary). 160 CONJUGATES OF A PERRON NUMBER AND ERDOS-TUR} AN¶ APPROACH Let us recall the Conditions of Parry [Bl] [Fr1] [Lo] [Pa].
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