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Author Verger-Gaugry, Jean-Louis
Source CiteSeerX
Content type Text
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Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Nyi Expansion ♦ Algebraic Number ♦ Class C1 ♦ Thue-siegel-roth Theorem ♦ New Classification Relies ♦ Maximal Asymptotic Quotient ♦ Salem Number ♦ Gap Value ♦ Mahler Measure ♦ New Classification ♦ Gappy Power Series ♦ Approximation Theorem ♦ Liouville Inequality ♦ Recent Generalization ♦ Natural Way ♦ Bertrand-mathis Classification
Abstract Let β> 1 be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi β-expansion dβ(1) of unity which controls the set Zβ of β-integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in dβ(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 (j) i which we compare to Bertrand-Mathis’s classification with classes C1 to C5 (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 dβ(1). As a corollary, all Salem numbers are in the class C1 ∪ Q (1) 0 ∪ Q(2) 0 ∪ Q(3) 0; this result is also directly proved using a recent generalization of the Thue-Siegel-Roth Theorem given by Corvaja.
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study
Learning Resource Type Article
Publisher Date 2006-01-01