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Archiv der Mathematik

, Volume 70, Issue 6, pp 455–463 | Cite as

A central limit theorem related to decimal and continued fraction expansion

  • Christian Faivre

Abstract.

For every irrational number \( x\in [0,1] \) and integer \( n\ge 1 \), we denote by \( k_n(x) \) the exact number of partial quotients of x which can be obtained from \( d_n(x) \) and \( e_n(x) \), the two consecutive n-decimal approximations of x. G. Lochs has proved that for almost all x, with respect to the Lebesgue measure ¶¶\( \lim \limits _{n\to \infty}{k_n(x)\over n}={6\,\log \,2\,\log 10\over \pi ^2} \). ¶¶In this paper the author proves that a central limit theorem holds for the sequence \( (k_n) \) i.e. more precisely¶¶\( m\left \{x\in [0,1];\ {k_n(x)-na\over \sigma \sqrt {n}}\le z\right \}\to {1\over \sqrt {2\pi }}\int\limits _{-\infty }^z e^{-t^2/2}\,dt \),¶¶ for some constant \( \sigma \ge 0 \), where \( a=6\,\log 2\,\log 10/\pi ^2 \) and m the Lebesgue measure.

Keywords

Limit Theorem Lebesgue Measure Central Limit Central Limit Theorem Exact Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel 1998

Authors and Affiliations

  • Christian Faivre
    • 1
  1. 1.Centre de Mathématiques et Informatique de l'Université de Provence, 39, rue Joliot Curie, F-13453 Marseille Cédex 13, FranceFR

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