Slow continued fractions, transducers, and the Serret theorem

JOURNAL OF NUMBER THEORY

卷:185

Article

Panti, Giovanni¹ 

[1] Univ Udine, Dept Math, Via Sci 206, I-33100 Udine, Italy

关键词: Continued fractions;    Gauss maps;    Tail property;    Extended modular group;    Transducers;   

DOI  :  10.1016/j.jnt.2017.08.034

来源: Elsevier

【 摘 要 】

A basic result in the elementary theory of continued fractions says that two real numbers share the same tail in their continued fraction expansions if they belong to the same orbit under the projective action of PGL(2)Z. This result was first formulated in Serret's Cours d'algebre superieure, so we'll refer to it as to the Serret theorem. Notwithstanding the abundance of continued fraction algorithms in the literature, a uniform treatment of the Serret result seems missing. In this paper we show that there are finitely many possibilities for the groups Sigma <= PGL(2) Z generated by the branches of the Gauss maps in a large family of algorithms, and that each Sigma-equivalence class of reals is partitioned in finitely many tail-equivalence classes, whose number we bound. Our approach is through the finite-state transducers that relate Gauss maps to each other. They constitute opfibrations of the Schreier graphs of the groups, and their synchronizability-which may or may not hold-assures the a.e. validity of the Serret theorem. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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