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/

 E. Bombieri and J. Mueller, Remarks on the approximation to an algebraic number by algebraic numbers, Michigan Math. J. 33 (1986), 83–93. | MR 817911 | Zbl 0593.10031

 Y. Bugeaud, “Approximation by Algebraic Numbers”, Cambridge Tracts in Mathematics 160, Cambridge University Press, 2004. | MR 2136100 | Zbl 1055.11002

 Y. Bugeaud and M. Laurent, Exponents of Diophantine approximation and Sturmian continued fractions, Ann. Inst. Fourier (Grenoble) 55 (2005), 773–804. | Numdam | MR 2149403 | Zbl 1155.11333

 Y. Bugeaud and M. Laurent, On exponents of homogeneous and inhomogeneous Diophantine approximation, Mosc. Math. J. 5 (2005), 747–766. | MR 2266457 | Zbl 1119.11039

 Y. Bugeaud and O. Teulié, Approximation d’un nombre réel par des nombres algébriques de degré donné, Acta Arith. 93 (2000), 77–86. | MR 1760090

 J. W. S. Cassels, “An Introduction to the Geometry of Numbers”, Springer Verlag, 1997. | MR 1434478

 H. Davenport and W. M. Schmidt, Approximation to real numbers by quadratic irrationals, Acta Arith. 13 (1967), 169–176. | MR 219476 | Zbl 0155.09503

 H. Davenport and W. M. Schmidt, A theorem on linear forms, Acta Arith. 14 (1967/1968), 209–223. | MR 225728 | Zbl 0179.07303

 H. Davenport and W. M. Schmidt, Approximation to real numbers by algebraic integers, Acta Arith. 15 (1969), 393–416. | MR 246822 | Zbl 0186.08603

 H. Davenport and W. M. Schmidt, Dirichlet’s theorem on Diophantine approximation II, Acta Arith. 16 (1970), 413–423. | MR 279040 | Zbl 0201.05501

 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 1915209 | Zbl 1026.11060

 A. Ya. Khintchine, Über eine Klasse linearer diophantischer Approximationen, Rend. Circ. Mat. Palermo 50 (1926), 170–195. | JFM 52.0183.01

 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 65.0180.01 | MR 845

 K. Mahler, Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II, J. Reine Angew. Math. 166 (1932), 118–150. | JFM 58.0207.01 | MR 1581302

 K. F. Roth, Rational approximations to algebraic numbers, Matematika 2 (1955), 1–20; corrigendum, 168. | MR 72182 | Zbl 0064.28501

 D. Roy, Approximation simultanée d’un nombre et son carré, C. R. Acad. Sci. Paris 336 (2003), 1–6. | MR 1968892

 D. Roy, Approximation to real numbers by cubic algebraic numbers, I, Proc. London Math. Soc. 88 (2004), 42–62. | MR 2018957 | Zbl 1035.11028

 D. Roy, Approximation to real numbers by cubic algebraic numbers, II, Ann. of Math. 158 (2003), 1081–1087. | MR 2031862 | Zbl 1044.11061

 D. Roy and M. Waldschmidt, Diophantine approximation by conjugate algebraic integers, Compositio Math. 140 (2004), 593–612. | MR 2041771 | Zbl 1055.11043

 W. M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math. 125 (1970), 189–201. | MR 268129 | Zbl 0205.06702

 W. M. Schmidt, Linearformen mit algebraischen Koeffizienten. II, Math. Ann. 191 (1971), 1–20. | MR 308062 | Zbl 0198.07103

 W. M. Schmidt, “Approximation to Algebraic Numbers”, Monographie de l’Enseignement Mathématique 19, Genève, 1971. | MR 327672 | Zbl 0226.10033

 W. M. Schmidt, “Diophantine Approximation”, Lecture Notes in Math. 785, Springer, Berlin, 1980. | MR 568710 | Zbl 0421.10019

 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. | MR 142510 | Zbl 0097.03503