Research output: Contribution to journal › Article › Scientific › peer-review

Denote by p n /q n ,n=1,2,3,…, pn/qn,n=1,2,3,…,
the sequence of continued fraction convergents of a real irrational number x x
. Define the sequence of approximation coefficients by θ n (x):=q n |q n x−p n |,n=1,2,3,… θn(x):=qn|qnx−pn|,n=1,2,3,…
. In the case of regular continued fractions the six possible patterns of three consecutive approximation coefficients, such as θ n−1 <θ n <θ n+1 θn−1<θn<θn+1
, occur for almost all x x
with only two different asymptotic frequencies. In this paper it is shown how these asymptotic frequencies can be determined for two other semi-regular cases. It appears that the *optimal continued fraction* has a similar distribution of only two asymptotic frequencies, albeit with different values. The six different values that are found in the case of the *nearest integer continued fraction* will show to be closely related to those of the optimal continued fraction.

Original language | English |
---|---|

Pages (from-to) | 285-317 |

Number of pages | 33 |

Journal | Tohoku Mathematical Journal |

Volume | 70 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2018 |

- Continued fractions, Metric theory

ID: 45772096