Sur la somme des quotients partiels du développement en fraction continue

D. Barbolosi; C. Faivre

Colloquium Mathematicae (2001)

  • Volume: 89, Issue: 2, page 159-167
  • ISSN: 0010-1354

Abstract

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Let [0;a₁(x),a₂(x),…] be the regular continued fraction expansion of an irrational x ∈ [0,1]. We prove mainly that, for α > 0, β ≥ 0 and for almost all x ∈ [0,1], l i m n → ∞ ( a ⁿ ₁ ( x ) + … + a ⁿ ₙ ( x ) ) / n l o g n = α / l o g 2 if α < 1 and β ≥ 0, l i m n → ∞ ( a ⁿ ₁ ( x ) + … + a ⁿ ₙ ( x ) ) / n l o g n = 1 / l o g 2 if α = 1 and β < 1, and, if α > 1 or α = 1 and β >1, l i m   i n f n → ∞ ( a ⁿ ₁ ( x ) + … + a ⁿ ₙ ( x ) ) / n l o g n = 1 / l o g 2 , l i m   s u p n → ∞ ( a ⁿ ₁ ( x ) + … + a ⁿ ₙ ( x ) ) / n l o g n = ∞ , where a ⁿ i ( x ) = a i ( x ) if a i ( x ) ≤ n α l o g β n and a ⁿ i ( x ) = 0 otherwise, for all i ∈ 1,…,n.

How to cite

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D. Barbolosi, and C. Faivre. "Sur la somme des quotients partiels du développement en fraction continue." Colloquium Mathematicae 89.2 (2001): 159-167. <http://eudml.org/doc/284290>.

@article{D2001,
author = {D. Barbolosi, C. Faivre},
journal = {Colloquium Mathematicae},
keywords = {sums of partial quotients; exponentially mixing; strong law of large numbers; regular continued fraction expansion},
language = {fre},
number = {2},
pages = {159-167},
title = {Sur la somme des quotients partiels du développement en fraction continue},
url = {http://eudml.org/doc/284290},
volume = {89},
year = {2001},
}

TY - JOUR
AU - D. Barbolosi
AU - C. Faivre
TI - Sur la somme des quotients partiels du développement en fraction continue
JO - Colloquium Mathematicae
PY - 2001
VL - 89
IS - 2
SP - 159
EP - 167
LA - fre
KW - sums of partial quotients; exponentially mixing; strong law of large numbers; regular continued fraction expansion
UR - http://eudml.org/doc/284290
ER -

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