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/

[1] 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

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

[3] 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

[4] 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

[5] 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

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

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

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

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

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

[11] 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

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

[13] 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

[14] 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

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

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

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

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

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

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

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

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

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

[24] 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.

[25] E. Wirsing, Approximation mit algebraischen Zahlen beschränkten Grades, J. Reine Angew. Math. 206 (1961), 67–77. | MR 142510 | Zbl 0097.03503