On gaps in Rényi $\beta$-expansions of unity for $\beta >1$ an algebraic number

Verger-Gaugry, Jean-Louis

Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2565-2579.

Soit $\beta >1$ un nombre algébrique. Nous étudions les plages de zéros (“lacunes”) dans le $\beta$-développement de Rényi  $d_{\beta }\left(1\right)$ de l’unité qui contrôle l’ensemble ${\mathbb{Z}}_{\beta }$ des $\beta$-entiers. En utilisant une version de l’inégalité de Liouville qui étend des théorèmes d’approximation de Mahler et de Güting, on montre que les plages de zéros dans $d_{\beta }\left(1\right)$ présentent une “lacunarité” asymptotiquement bornée supérieurement par  $log\left(\mathrm{M}\right(\beta \left)\right)/log\left(\beta \right)$, où  $\mathrm{M}\left(\beta \right)$  est la mesure de Mahler de  $\beta$. La preuve de ce résultat fournit de manière naturelle une nouvelle classification des nombres algébriques $>1$ en classes appelées Q${}_i^{\left(j\right)}$ que nous comparons à la classification de Bertrand-Mathis avec les classes C${}_1$ à C${}_5$ (reportée dans un article de Blanchard). Cette nouvelle classification repose sur la valeur asymptotique maximale du “quotient de lacune” de la série “lacunaire” associée à $d_{\beta }\left(1\right)$. Comme corollaire, tous les nombres de Salem sont dans la classe C${}_1\cup$Q${}_0^{\left(1\right)}\cup$ Q${}_0^{\left(2\right)}\cup$ Q${}_0^{\left(3\right)}$ ; ce résultat est également obtenu par un théorème récent qui généralise le théorème de Thue-Siegel-Roth donné par Corvaja.

Let $\beta >1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $\beta$-expansion  $d_{\beta }\left(1\right)$ of unity which controls the set ${\mathbb{Z}}_{\beta }$ of $\beta$-integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in $d_{\beta }\left(1\right)$ are shown to exhibit a “gappiness” asymptotically bounded above by  $log\left(\mathrm{M}\right(\beta \left)\right)/log\left(\beta \right)$, where  $\mathrm{M}\left(\beta \right)$  is the Mahler measure of  $\beta$. The proof of this result provides in a natural way a new classification of algebraic numbers $>1$ with classes called Q${}_i^{\left(j\right)}$ which we compare to Bertrand-Mathis’s classification with classes C${}_1$ to C${}_5$ (reported in an article by Blanchard). This new classification relies on the maximal asymptotic “quotient of the gap” value of the “gappy” power series associated with $d_{\beta }\left(1\right)$. As a corollary, all Salem numbers are in the class C${}_1\cup$Q${}_0^{\left(1\right)}\cup$ Q${}_0^{\left(2\right)}\cup$ Q${}_0^{\left(3\right)}$ ; this result is also directly proved using a recent generalization of the Thue-Siegel-Roth Theorem given by Corvaja.

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