Approximation of complex algebraic numbers by algebraic numbers of bounded degree
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 2, pp. 333-368.

To measure how well a given complex number $\xi$ can be approximated by algebraic numbers of degree at most $n$ one may use the quantities ${w}_{n}\left(\xi \right)$ and ${w}_{n}^{*}\left(\xi \right)$ introduced by Mahler and Koksma, respectively. The values of ${w}_{n}\left(\xi \right)$ and ${w}_{n}^{*}\left(\xi \right)$ have been computed for real algebraic numbers $\xi$, but up to now not for complex, non-real algebraic numbers $\xi$. In this paper we compute ${w}_{n}\left(\xi \right)$, ${w}_{n}^{*}\left(\xi \right)$ for all positive integers $n$ and algebraic numbers $\xi \in ℂ\setminus ℝ$, except for those pairs $\left(n,\xi \right)$ such that $n$ is even, $n\ge 6$ and $n+3\le deg\xi \le 2n-2$. It is known that every real algebraic number of degree $>n$ has the same values for ${w}_{n}$ and ${w}_{n}^{*}$ as almost every real number. Our results imply that for every positive even integer $n$ there are complex algebraic numbers $\xi$ of degree $>n$ which are unusually well approximable by algebraic numbers of degree at most $n$, i.e., have larger values for ${w}_{n}$ and ${w}_{n}^{*}$ than almost all complex numbers. We consider also the approximation of complex non-real algebraic numbers $\xi$ by algebraic integers, and show that if $\xi$ is unusually well approximable by algebraic numbers of degree at most $n$ then it is unusually badly approximable by algebraic integers of degree at most $n+1$. By means of Schmidt’s Subspace Theorem we reduce the approximation problem to compute ${w}_{n}\left(\xi \right)$, ${w}_{n}^{*}\left(\xi \right)$ to an algebraic problem which is trivial if $\xi$ is real but much harder if $\xi$ is not real. We give a partial solution to this problem.

Classification : 11J68
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Bugeaud, Yann; Evertse, Jan-Hendrik. Approximation of complex algebraic numbers by algebraic numbers of bounded degree. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 2, pp. 333-368. http://www.numdam.org/item/ASNSP_2009_5_8_2_333_0/

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