Approximation of complex algebraic numbers by algebraic numbers of bounded degree
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 2, p. 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
@article{ASNSP_2009_5_8_2_333_0,
author = {Bugeaud, Yann and Evertse, Jan-Hendrik},
title = {Approximation of complex algebraic numbers by algebraic numbers of bounded degree},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 8},
number = {2},
year = {2009},
pages = {333-368},
zbl = {1176.11031},
mrnumber = {2548250},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2009_5_8_2_333_0}
}
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, Serie 5, Volume 8 (2009) no. 2, pp. 333-368. http://www.numdam.org/item/ASNSP_2009_5_8_2_333_0/

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