Research Output per year

### Abstract

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 |
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Pages (from-to) | 285-317 |

Number of pages | 33 |

Journal | Tohoku Mathematical Journal |

Volume | 70 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2018 |

### Keywords

- Continued fractions
- Metric theory

## Fingerprint Dive into the research topics of 'Three consecutive approximation coefficients: Asymptotic frequencies in semi-regular cases'. Together they form a unique fingerprint.

## Research Output

- 1 Dissertation (TU Delft)

## Gaps, Frequencies and Spacial Limits of Continued Fraction Expansions

de Jonge, J., 2020, 133 p.Research output: Thesis › Dissertation (TU Delft)