Some metric properties of α-continued fractions

Hitoshi Nakada; Rie Natsui

doi:10.1016/S0022-314X(02)00008-2

Some metric properties of α-continued fractions

Hitoshi Nakada, Rie Natsui

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

8 Citations (Scopus)

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 max₁ ≤ n ≤ N C_n. Indeed, we prove that exist and have the same constant for almost every x.

Original language	English
Pages (from-to)	287-300
Number of pages	14
Journal	Journal of Number Theory
Volume	97
Issue number	2
DOIs	https://doi.org/10.1016/S0022-314X(02)00008-2
Publication status	Published - 2002 Dec 1
Externally published	Yes

Keywords

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

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/S0022-314X(02)00008-2

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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 Cn ≥ 1 and εn = ±1 for n ≥ 1. We prove the Borel-Bernstein theorem for α-continued fractions and also discuss some metrical properties related to max1 ≤ n ≤ N Cn. Indeed, we prove that exist and have the same constant for almost every x.",

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AU - Natsui, Rie

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N2 - 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 Cn ≥ 1 and εn = ±1 for n ≥ 1. We prove the Borel-Bernstein theorem for α-continued fractions and also discuss some metrical properties related to max1 ≤ n ≤ N Cn. Indeed, we prove that exist and have the same constant for almost every x.

AB - 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 Cn ≥ 1 and εn = ±1 for n ≥ 1. We prove the Borel-Bernstein theorem for α-continued fractions and also discuss some metrical properties related to max1 ≤ n ≤ N Cn. Indeed, we prove that exist and have the same constant for almost every x.

KW - Borel-Bernstein theorem

KW - Maxima of continued fraction coefficients

KW - α-Continued fractions

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SP - 287

EP - 300

JO - Journal of Number Theory

JF - Journal of Number Theory

IS - 2

ER -

Some metric properties of α-continued fractions

Abstract

Keywords

ASJC Scopus subject areas

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