Sur la somme des quotients partiels du développement en fraction continue
Volume 89 / 2001
Abstract
Let $[0;a_{1}(x),a_{2}(x),\ldots ]$ be the regular continued fraction expansion of an irrational $x\in[ 0,1]$. We prove mainly that, for $\alpha >0$, $\beta \geq 0$ and for almost all $x\in [0,1]$, $$ \lim _{n\to \infty }\frac{a_{1}^{n}(x)+\ldots +a_{n}^{n}(x)}{n\log n}= \cases{ {\alpha }/\!\log 2&{\rm if}\ \alpha <1\ {\rm and}\ \beta \geq 0,\cr {1}/\!\log 2&{\rm if}\ \alpha =1\ {\rm and}\ \beta<1,\cr} $$ and, if $\alpha >1$ or $\alpha =1\ {\rm and }\ \beta >1$, $$\eqalign{ &\liminf _{n\to \infty }\frac{a_{1}^{n}(x)+\ldots +a_{n}^{n}(x)}{n\log n}=\frac{1}{\log 2},\cr &\limsup _{n\to \infty}\frac{a_{1}^{n}(x)+\ldots +a_{n}^{n}(x)}{n\log n} =\infty,\cr}$$ where $a_{i}^{n}(x)=a_{i}(x)$ if $a_{i}(x)\leq n^{\alpha }\log ^{\beta }n$ and $a_{i}^{n}(x)=0$ otherwise, for all $i\in \{ 1,\ldots ,n\}$.
