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 ξ can be approximated by algebraic numbers of degree at most n one may use the quantities w n (ξ) and w n * (ξ) introduced by Mahler and Koksma, respectively. The values of w n (ξ) and w n * (ξ) have been computed for real algebraic numbers ξ, but up to now not for complex, non-real algebraic numbers ξ. In this paper we compute w n (ξ), w n * (ξ) for all positive integers n and algebraic numbers ξ, except for those pairs (n,ξ) such that n is even, n6 and n+3degξ2n-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 ξ 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 ξ by algebraic integers, and show that if ξ 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 (ξ), w n * (ξ) to an algebraic problem which is trivial if ξ is real but much harder if ξ 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},
     mrnumber = {2548250},
     zbl = {1176.11031},
     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, Série 5, Tome 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 | Zbl

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

[3] 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

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

[5] 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

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

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

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

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

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

[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 | Zbl

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

[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 | MR

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

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

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

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

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

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

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

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

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

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

[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. | EuDML | MR | Zbl