On gaps in Rényi β -expansions of unity for β > 1 an algebraic number

Jean-Louis Verger-Gaugry[1]

  • [1] Université de Grenoble I Institut Fourier UMR CNRS 5582 BP 74 - Domaine Universitaire, 38402 Saint-Martin d’Hères (France)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 7, page 2565-2579
  • ISSN: 0373-0956

Abstract

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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 ∪ 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.

How to cite

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Verger-Gaugry, Jean-Louis. "On gaps in Rényi $\beta $-expansions of unity for $\beta &gt; 1$ an algebraic number." Annales de l’institut Fourier 56.7 (2006): 2565-2579. <http://eudml.org/doc/10214>.

@article{Verger2006,
abstract = {Let $\beta &gt; 1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $\beta $-expansion  $d_\{\beta \}(1)$ 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 $d_\{\beta \}(1)$ are shown to exhibit a “gappiness” asymptotically bounded above by  $\log (\{\rm M\}(\beta ))/\log (\beta )$, where  $\{\rm M\}(\beta )$  is the Mahler measure of  $\beta $. The proof of this result provides in a natural way a new classification of algebraic numbers $&gt; 1$ 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 $d_\{\beta \}(1)$. 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.},
affiliation = {Université de Grenoble I Institut Fourier UMR CNRS 5582 BP 74 - Domaine Universitaire, 38402 Saint-Martin d’Hères (France)},
author = {Verger-Gaugry, Jean-Louis},
journal = {Annales de l’institut Fourier},
keywords = {Beta-integer; beta-numeration; PV number; Salem number; Perron number; Mahler measure; Diophantine approximation; Mahler’s series; mathematical quasicrystal; beta expansions; Pisot number},
language = {eng},
number = {7},
pages = {2565-2579},
publisher = {Association des Annales de l’institut Fourier},
title = {On gaps in Rényi $\beta $-expansions of unity for $\beta &gt; 1$ an algebraic number},
url = {http://eudml.org/doc/10214},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Verger-Gaugry, Jean-Louis
TI - On gaps in Rényi $\beta $-expansions of unity for $\beta &gt; 1$ an algebraic number
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 7
SP - 2565
EP - 2579
AB - Let $\beta &gt; 1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $\beta $-expansion  $d_{\beta }(1)$ 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 $d_{\beta }(1)$ are shown to exhibit a “gappiness” asymptotically bounded above by  $\log ({\rm M}(\beta ))/\log (\beta )$, where  ${\rm M}(\beta )$  is the Mahler measure of  $\beta $. The proof of this result provides in a natural way a new classification of algebraic numbers $&gt; 1$ 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 $d_{\beta }(1)$. 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.
LA - eng
KW - Beta-integer; beta-numeration; PV number; Salem number; Perron number; Mahler measure; Diophantine approximation; Mahler’s series; mathematical quasicrystal; beta expansions; Pisot number
UR - http://eudml.org/doc/10214
ER -

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