Solution of Two Problems of Mahler and Mendès France
نویسندگان
چکیده
Abstract. About fifty years ago Mahler [M] proved that if α > 1 is rational but not an integer and if 0 < l < 1, then the fractional part of α is > l apart from a finite set of integers n depending on α and l. Answering completely a question of Mahler, we show (Thm. 1) that the same conclusion holds for all algebraic numbers which are not d-th roots of Pisot numbers. By related methods we also answer a question of Mendès France, characterizing completely the quadratic irrationals α such that the continued fraction for α has period length tending to infinity (Thm. 2).
منابع مشابه
Function fields in positive characteristic: Expansions and Cobham’s theorem
In the vein of Christol, Kamae, Mendès France and Rauzy, we consider the analogue of a problem of Mahler for rational functions in positive characteristic. To solve this question, we prove an extension of Cobham’s theorem for quasi-automatic functions and use the recent generalization of Christol’s theorem obtained by Kedlaya. 2008 Elsevier Inc. All rights reserved.
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GET ENST Bretagne / Département LUSSI – CNRS UMR 2872 Technopôle de Brest Iroise CS 83818, 29238 Brest Cedex, France {prenom.nom}@enst-bretagne.fr Laboratoire ERIC Université Lumière Lyon 2 5 avenue Pierre Mendès-France, 69676 Bron Cedex, France [email protected] ∗∗∗Service de Mathématiques Appliquées, Faculté de Droit, d’Economie et de Finance, Université du Luxembourg, 162a, avenue de la ...
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