Sur la somme des quotients partiels du développement en fraction continue

D. Barbolosi; C. Faivre

D. Barbolosi ; C. Faivre

Colloquium Mathematicae, Tome 89 (2001), p. 159-167 / Harvested from The Polish Digital Mathematics Library

Résumé

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], $l i m_{n \to \infty} (a ⁿ ₁ (x) + \dots + a ⁿ ₙ (x)) / n l o g n = α / l o g 2$ if α < 1 and β ≥ 0, $l i m_{n \to \infty} (a ⁿ ₁ (x) + \dots + a ⁿ ₙ (x)) / n l o g n = 1 / l o g 2$ if α = 1 and β < 1, and, if α > 1 or α = 1 and β >1, $l i m i n f_{n \to \infty} (a ⁿ ₁ (x) + \dots + a ⁿ ₙ (x)) / n l o g n = 1 / l o g 2$ , $l i m s u p_{n \to \infty} (a ⁿ ₁ (x) + \dots + a ⁿ ₙ (x)) / n l o g n = \infty$ , where $a ⁿ_{i} (x) = a_{i} (x)$ if $a_{i} (x) \leq n^{α} l o g^{β} n$ and $a ⁿ_{i} (x) = 0$ otherwise, for all i ∈ 1,…,n.

Publié le : 2001-01-01

Zbl 1004.11046

EUDML-ID : urn:eudml:doc:284290

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     title = {Sur la somme des quotients partiels du d\'eveloppement en fraction continue},
     journal = {Colloquium Mathematicae},
     volume = {89},
     year = {2001},
     pages = {159-167},
     zbl = {1004.11046},
     language = {fra},
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D. Barbolosi; C. Faivre. Sur la somme des quotients partiels du développement en fraction continue. Colloquium Mathematicae, Tome 89 (2001) pp. 159-167. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-2-1/