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Sur la somme des quotients partiels du développement en fraction continue

D. Barbolosi, C. Faivre (2001)

Colloquium Mathematicae

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.

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