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On the dichotomy of Perron numbers and beta-conjugates

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Abstract

Let β > 1 be an algebraic number. A general definition of a beta-conjugate of β is proposed with respect to the analytical function \({f_{\beta}(z) =-1 + \sum_{i \geq 1} t_i z^i}\) associated with the Rényi β-expansion d β (1) = 0.t 1 t 2 . . . of unity. From Szegő’s Theorem, we study the dichotomy problem for f β (z), in particular for β 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 Erdős–Turán 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 \({{\rm C}_1 \cup \,{\rm C}_2 \cup \,{\rm C}_3}\) after the classification of Blanchard/Bertrand-Mathis.

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  1. Institut Fourier, CNRS UMR 5582, Université de Grenoble I, BP 74, 38402, Saint-Martin d’Hères, France

    Jean-Louis Verger-Gaugry

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Correspondence to Jean-Louis Verger-Gaugry.

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Work supported by ACINIM 2004-154 “Numeration”.

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Verger-Gaugry, JL. On the dichotomy of Perron numbers and beta-conjugates. Monatsh Math 155, 277–299 (2008). https://doi.org/10.1007/s00605-008-0002-1

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