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

Verger-Gaugry, Jean-Louis

On gaps in Rényi

β

-expansions of unity for

β > 1

an algebraic number
[Sur les lacunes du

β

-développement de Rényi de l’unité pour

β > 1

un nombre algébrique]

Verger-Gaugry, Jean-Louis

Annales de l'Institut Fourier, Tome 56 (2006), p. 2565-2579 / Harvested from Numdam

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Résumé
Abstract

Soit $β > 1$ un nombre algébrique. Nous étudions les plages de zéros (“lacunes”) dans le $β$ -développement de Rényi $d_{β} (1)$ de l’unité qui contrôle l’ensemble $ℤ_{β}$ des $β$ -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 $d_{β} (1)$ présentent une “lacunarité” asymptotiquement bornée supérieurement par $log (M (β)) / log (β)$ , où $M (β)$ est la mesure de Mahler de $β$ . La preuve de ce résultat fournit de manière naturelle une nouvelle classification des nombres algébriques $> 1$ en classes appelées Q $_{i}^{(j)}$ 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 $d_{β} (1)$ . Comme corollaire, tous les nombres de Salem sont dans la classe C $_{1} \cup$ Q $_{0}^{(1)} \cup$ Q $_{0}^{(2)} \cup$ Q $_{0}^{(3)}$ ; ce résultat est également obtenu par un théorème récent qui généralise le théorème de Thue-Siegel-Roth donné par Corvaja.

Let $β > 1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $β$ -expansion $d_{β} (1)$ of unity which controls the set $ℤ_{β}$ of $β$ -integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in $d_{β} (1)$ are shown to exhibit a “gappiness” asymptotically bounded above by $log (M (β)) / log (β)$ , where $M (β)$ is the Mahler measure of $β$ . The proof of this result provides in a natural way a new classification of algebraic numbers $> 1$ with classes called Q $_{i}^{(j)}$ 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 $d_{β} (1)$ . As a corollary, all Salem numbers are in the class C $_{1} \cup$ Q $_{0}^{(1)} \cup$ Q $_{0}^{(2)} \cup$ Q $_{0}^{(3)}$ ; this result is also directly proved using a recent generalization of the Thue-Siegel-Roth Theorem given by Corvaja.

Publié le : 2006-01-01

MR 2290791 | Zbl pre05176579 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2250
Classification: 11B05, 11Jxx, 11J68, 11R06, 52C23
Mots clés: Beta-entier, beta-numération, nombre de Pisot, nombre de Salem, nombre de Perron, mesure de Mahler, approximation Diophantienne, série de Mahler, quasicristal mathématique

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     author = {Verger-Gaugry, Jean-Louis},
     title = {On gaps in R\'enyi $\beta $-expansions of unity for $\beta > 1$ an algebraic number},
     journal = {Annales de l'Institut Fourier},
     volume = {56},
     year = {2006},
     pages = {2565-2579},
     doi = {10.5802/aif.2250},
     zbl = {pre05176579},
     mrnumber = {2290791},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2006__56_7_2565_0}
}

Verger-Gaugry, Jean-Louis. On gaps in Rényi $\beta $-expansions of unity for $\beta > 1$ an algebraic number. Annales de l'Institut Fourier, Tome 56 (2006) pp. 2565-2579. doi : 10.5802/aif.2250. http://gdmltest.u-ga.fr/item/AIF_2006__56_7_2565_0/

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