### Abstract

In this paper we prove a renewal-type limit theorem. Given α \in (0,1)\backslash \mathbb {Q} and R>0, let qnR be the first denominator of the convergents of which exceeds R. The main result in the paper is that the ratio qnR/R has a limiting distribution as R tends to infinity. The existence of the limiting distribution uses mixing of a special flow over the natural extension of the Gauss map.

Original language | English (US) |
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Pages (from-to) | 643-655 |

Number of pages | 13 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2008 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

Sinai, Y. G., & Ulcigrai, C. (2008). Renewal-type limit theorem for the Gauss map and continued fractions.

*Ergodic Theory and Dynamical Systems*,*28*(2), 643-655. https://doi.org/10.1017/S0143385707000466