On simultaneous rational approximation to a real number and its integral powers

Bugeaud, Yann

Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2165-2182.

Pour un entier strictement positif $n$ et un nombre réel $\xi$, on note ${\lambda }_n\left(\xi \right)$ le supremum des nombres réels $\lambda$ pour lesquels il existe des entiers $q$ arbitrairement grands tels que $\left|\right|q\xi \left|\right|,\left|\right|q{\xi }^2\left|\right|,...,\left|\right|q{\xi }^n\left|\right|$ sont tous inférieurs à $q^{-\lambda }$. Ici, $\left|\right|·\left|\right|$ désigne la distance à l’entier le plus proche. Nous étudions l’ensemble des valeurs prises par la function ${\lambda }_n$ et, plus généralement, nous nous intéressons au spectre de $({\lambda }_1,...,{\lambda }_n,...)$. Nous formulons également plusieurs problèmes ouverts.

For a positive integer $n$ and a real number $\xi$, let ${\lambda }_n\left(\xi \right)$ denote the supremum of the real numbers $\lambda$ such that there are arbitrarily large positive integers $q$ such that $\left|\right|q\xi \left|\right|,\left|\right|q{\xi }^2\left|\right|,...,\left|\right|q{\xi }^n\left|\right|$ are all less than $q^{-\lambda }$. Here, $\left|\right|·\left|\right|$ denotes the distance to the nearest integer. We study the set of values taken by the function ${\lambda }_n$ and, more generally, we are concerned with the joint spectrum of $({\lambda }_1,...,{\lambda }_n,...)$. We further address several open problems.

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