Sur la somme des quotients partiels du développement en fraction continue

D. Barbolosi¹ ; C. Faivre²

¹ Faculté des Sciences et Techniques de St. Jerôme Service de Mathématiques Case 322 Avenue Escadrille Normandie-Niemen 13397 Marseille Cedex 20, France
² Centre de Mathématiques et Informatique de l'Université de Provence 39, rue Joliot Curie 13453 Marseille Cedex 13, France

Colloquium Mathematicum, Tome 89 (2001) no. 2, pp. 159-167

Résumé

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 $x\in [0,1]$, $$ \lim _{n\to \infty }\frac{a_{1}^{n}(x)+\ldots +a_{n}^{n}(x)}{n\log n}= \cases{ {\alpha }/\!\log 2{\rm if}\ \alpha 1\ {\rm and}\ \beta \geq 0,\cr {1}/\!\log 2{\rm if}\ \alpha =1\ {\rm and}\ \beta1,\cr} $$ 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 $i\in \{ 1,\ldots ,n\}$.

Détail

DOI : 10.4064/cm89-2-1

Mots-clés : ldots regular continued fraction expansion irrational prove mainly alpha beta geq almost lim infty frac ldots log cases alpha log alpha beta geq log alpha beta alpha alpha beta eqalign liminf infty frac ldots log frac log limsup infty frac ldots log infty where leq alpha log beta otherwise ldots