On the length of the continued fraction for values of quotients of power sums

Corvaja, Pietro; Zannier, Umberto

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 737-748.

En généralisant un résultat de Pourchet, nous démontrons que si $\alpha ,\beta$ sont deux sommes de puissances définies sur $\mathbb{Q}$, satisfaisant certaines conditions nécessaires, la longueur de la fraction continue pour $\alpha \left(n\right)/\beta \left(n\right)$ tend vers l’infini pour $n\to \infty$. On déduira ce résultat d’une inégalité de type Thue uniforme pour les approximations rationnelles des nombres de la forme $\alpha \left(n\right)/\beta \left(n\right)$.

Generalizing a result of Pourchet, we show that, if $\alpha ,\beta$ are power sums over $\mathbb{Q}$ satisfying suitable necessary assumptions, the length of the continued fraction for $\alpha \left(n\right)/\beta \left(n\right)$ tends to infinity as $n\to \infty$. This will be derived from a uniform Thue-type inequality for the rational approximations to the rational numbers $\alpha \left(n\right)/\beta \left(n\right)$, $n\in \mathbb{N}$.

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