On gaps in Rényi β-expansions of unity for β>1 an algebraic number
Annales de l'Institut Fourier, Volume 56 (2006) no. 7, pp. 2565-2579.

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 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 Q 0 (1) Q 0 (2) Q 0 (3) ; this result is also directly proved using a recent generalization of the Thue-Siegel-Roth Theorem given by Corvaja.

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 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 Q 0 (1) Q 0 (2) Q 0 (3)  ; ce résultat est également obtenu par un théorème récent qui généralise le théorème de Thue-Siegel-Roth donné par Corvaja.

DOI: 10.5802/aif.2250
Classification: 11B05,  11Jxx,  11J68,  11R06,  52C23
Keywords: Beta-integer, beta-numeration, PV number, Salem number, Perron number, Mahler measure, Diophantine approximation, Mahler’s series, mathematical quasicrystal
Verger-Gaugry, Jean-Louis 1

1 Université de Grenoble I Institut Fourier UMR CNRS 5582 BP 74 - Domaine Universitaire, 38402 Saint-Martin d’Hères (France)
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Verger-Gaugry, Jean-Louis. On gaps in Rényi $\beta $-expansions of unity for $\beta > 1$ an algebraic number. Annales de l'Institut Fourier, Volume 56 (2006) no. 7, pp. 2565-2579. doi : 10.5802/aif.2250. http://www.numdam.org/articles/10.5802/aif.2250/

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