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.

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
Bugeaud, Yann 1; Evertse, Jan-Hendrik 2

1 Université Louis Pasteur, U. F. R. de mathématiques, 7, rue René Descartes, 67084 Strasbourg (France)
2 Universiteit Leiden, Mathematisch Instituut, Postbus 9512, 2300 RA Leiden (The Netherlands)
@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},
pages = {333--368},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 8},
number = {2},
year = {2009},
zbl = {1176.11031},
mrnumber = {2548250},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2009_5_8_2_333_0/}
}
TY  - JOUR
AU  - Bugeaud, Yann
AU  - Evertse, Jan-Hendrik
TI  - Approximation of complex algebraic numbers by algebraic numbers of bounded degree
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2009
SP  - 333
EP  - 368
VL  - 8
IS  - 2
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2009_5_8_2_333_0/
LA  - en
ID  - ASNSP_2009_5_8_2_333_0
ER  - 
%0 Journal Article
%A Bugeaud, Yann
%A Evertse, Jan-Hendrik
%T Approximation of complex algebraic numbers by algebraic numbers of bounded degree
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2009
%P 333-368
%V 8
%N 2
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2009_5_8_2_333_0/
%G en
%F 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/

 E. Bombieri and J. Mueller, Remarks on the approximation to an algebraic number by algebraic numbers, Michigan Math. J. 33 (1986), 83–93. | MR | Zbl

 Y. Bugeaud, “Approximation by Algebraic Numbers”, Cambridge Tracts in Mathematics 160, Cambridge University Press, 2004. | MR | Zbl

 Y. Bugeaud and M. Laurent, Exponents of Diophantine approximation and Sturmian continued fractions, Ann. Inst. Fourier (Grenoble) 55 (2005), 773–804. | EuDML | Numdam | MR | Zbl

 Y. Bugeaud and M. Laurent, On exponents of homogeneous and inhomogeneous Diophantine approximation, Mosc. Math. J. 5 (2005), 747–766. | MR | Zbl

 Y. Bugeaud and O. Teulié, Approximation d’un nombre réel par des nombres algébriques de degré donné, Acta Arith. 93 (2000), 77–86. | EuDML | MR

 J. W. S. Cassels, “An Introduction to the Geometry of Numbers”, Springer Verlag, 1997. | MR

 H. Davenport and W. M. Schmidt, Approximation to real numbers by quadratic irrationals, Acta Arith. 13 (1967), 169–176. | EuDML | MR | Zbl

 H. Davenport and W. M. Schmidt, A theorem on linear forms, Acta Arith. 14 (1967/1968), 209–223. | EuDML | MR | Zbl

 H. Davenport and W. M. Schmidt, Approximation to real numbers by algebraic integers, Acta Arith. 15 (1969), 393–416. | EuDML | MR | Zbl

 H. Davenport and W. M. Schmidt, Dirichlet’s theorem on Diophantine approximation II, Acta Arith. 16 (1970), 413–423. | EuDML | MR | Zbl

 J.-H. Evertse and H.P. Schlickewei, A quantitative version of the Absolute Parametric Subspace Theorem, J. Reine Angew. Math. 548 (2002), 21–127. | MR | Zbl

 A. Ya. Khintchine, Über eine Klasse linearer diophantischer Approximationen, Rend. Circ. Mat. Palermo 50 (1926), 170–195. | JFM

 J. F. Koksma, Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen, Monatsh. Math. Phys. 48 (1939), 176–189. | JFM | MR

 K. Mahler, Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II, J. Reine Angew. Math. 166 (1932), 118–150. | EuDML | JFM | MR

 K. F. Roth, Rational approximations to algebraic numbers, Matematika 2 (1955), 1–20; corrigendum, 168. | MR | Zbl

 D. Roy, Approximation simultanée d’un nombre et son carré, C. R. Acad. Sci. Paris 336 (2003), 1–6. | MR

 D. Roy, Approximation to real numbers by cubic algebraic numbers, I, Proc. London Math. Soc. 88 (2004), 42–62. | MR | Zbl

 D. Roy, Approximation to real numbers by cubic algebraic numbers, II, Ann. of Math. 158 (2003), 1081–1087. | MR | Zbl

 D. Roy and M. Waldschmidt, Diophantine approximation by conjugate algebraic integers, Compositio Math. 140 (2004), 593–612. | MR | Zbl

 W. M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math. 125 (1970), 189–201. | MR | Zbl

 W. M. Schmidt, Linearformen mit algebraischen Koeffizienten. II, Math. Ann. 191 (1971), 1–20. | EuDML | MR | Zbl

 W. M. Schmidt, “Approximation to Algebraic Numbers”, Monographie de l’Enseignement Mathématique 19, Genève, 1971. | MR | Zbl

 W. M. Schmidt, “Diophantine Approximation”, Lecture Notes in Math. 785, Springer, Berlin, 1980. | MR | Zbl

 V. G. Sprindžuk, “Mahler’s Problem in Metric Number Theory”, Izdat. “Nauka i Tehnika” , Minsk, 1967 (in Russian). English translation by B. Volkmann, Translations of Mathematical Monographs, Vol. 25, American Mathematical Society, Providence, R.I., 1969.

 E. Wirsing, Approximation mit algebraischen Zahlen beschränkten Grades, J. Reine Angew. Math. 206 (1961), 67–77. | EuDML | MR | Zbl