A central limit theorem related to decimal and continued fraction expansion

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.

Authors and Affiliations

1. Centre de Mathématiques et Informatique de l'Université de Provence, 39, rue Joliot Curie, F-13453 Marseille Cédex 13, France, , , , , , FR

   Christian Faivre

Received: 22.4.1997

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