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arXiv · 0903.2741

On a mixed problem in Diophantine approximation

Yann Bugeaud·Bernard De Mathan
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Abstract

Let $d$ be a positive integer. Let $p$ be a prime number. Let $α$ be a real algebraic number of degree $d+1$. We establish that there exist a positive constant $c$ and infinitely many algebraic numbers $ξ$ of degree $d$ such that $|α- ξ| \cdot \min\{|\Norm(ξ)|_p,1\} < c H(ξ)^{-d-1} (\log 3 H(ξ))^{-1/d}$. Here, $H(ξ)$ and $\Norm(ξ)$ denote the na{\"ı}ve height of $ξ$ and its norm, respectively. This extends an earlier result of de Mathan and Teulié that deals with the case $d=1$.

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Yann Bugeaud, Bernard De Mathan. 2009-03-16. On a mixed problem in Diophantine approximation. https://doi.org/10.4064/aa139-1-6

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