Approximation of complex algebraic numbers by algebraic numbers of bounded degree
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 8 (2009) no. 2, p. 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.
@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, Série 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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