On a mixed problem in Diophantine approximation
Résumé
Let $d$ be a positive integer. Let $p$ be a prime number. Let $\alpha$ be a real algebraic number of degree $d+1$. We establish that there exist a positive constant $c$ and infinitely many algebraic numbers $\xi$ of degree $d$ such that $|\alpha - \xi| \cdot \min\{|\Norm(\xi)|_p,1\} < c H(\xi)^{-d-1} \, (\log 3 H(\xi))^{-1/d}$. Here, $H(\xi)$ and $\Norm(\xi)$ denote the na\"\i ve height of $\xi$ and its norm, respectively. This extends an earlier result of de Mathan and Teulié that deals with the case $d=1$.
Domaines
Théorie des nombres [math.NT]
Origine : Fichiers produits par l'(les) auteur(s)
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