### Abstract

The α-continued fraction is a modification of the nearest integer continued fractions taking n as the integer part of y when y ε [n - 1 + α, n + α), instead of the nearest integer. For x εo[α - 1, α), we have the following α-continued fraction expansion: with C_{n} ≥ 1 and ε_{n} = ±1 for n ≥ 1. We prove the Borel-Bernstein theorem for α-continued fractions and also discuss some metrical properties related to max_{1} ≤ n ≤ N C_{n}. Indeed, we prove that exist and have the same constant for almost every x.

Original language | English |
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Pages (from-to) | 287-300 |

Number of pages | 14 |

Journal | Journal of Number Theory |

Volume | 97 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2002 Dec 1 |

### Keywords

- Borel-Bernstein theorem
- Maxima of continued fraction coefficients
- α-Continued fractions

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Nakada, H., & Natsui, R. (2002). Some metric properties of α-continued fractions.

*Journal of Number Theory*,*97*(2), 287-300. https://doi.org/10.1016/S0022-314X(02)00008-2