On the dichotomy of Perron numbers and beta-conjugates

Jean-Louis Verger-Gaugry (1)

1 IF - Institut Fourier

Let $\beta > 1$ be an algebraic number. A general definition of a beta-conjugate of f is proposed with respect to the analytical function $f_f(z) = −1 +\Sigma_{i \geq 1} t_i z^i$ associated with the Rényi f-expansion $d_f_{\beta}(1) = 0.t_1 t_2 . . .$ of unity. From Szeg¨o's Theorem, we study the dichotomy problem for $f_f(z)$, in particular for f 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 Erd˝os-Tur´an 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 C$_1 U $C$_2 U $C$_3$ after the classification of Blanchard/Bertrand-Mathis.

Mots clés: Perron number; Pisot number; Salem number; Erd\H{o}s-Turán's Theorem; numeration; Szeg\H{o}'s Theorem; uniform distribution; beta-shift; zeroes; betaconjugate

Domaines: Théorie des nombres [math.NT]

Jean-Louis Verger-Gaugry. On the dichotomy of Perron numbers and beta-conjugates. Monatshefte für Mathematik, 2008, 155, pp.277-299. ⟨10.1007/s00605-008-0002-1⟩. ⟨hal-00338280⟩