To measure how well a given complex number can be approximated by algebraic numbers of degree at most one may use the quantities and introduced by Mahler and Koksma, respectively. The values of and have been computed for real algebraic numbers , but up to now not for complex, non-real algebraic numbers . In this paper we compute , for all positive integers and algebraic numbers , except for those pairs such that is even, and . It is known that every real algebraic number of degree has the same values for and as almost every real number. Our results imply that for every positive even integer there are complex algebraic numbers of degree which are unusually well approximable by algebraic numbers of degree at most , i.e., have larger values for and than almost all complex numbers. We consider also the approximation of complex non-real algebraic numbers by algebraic integers, and show that if is unusually well approximable by algebraic numbers of degree at most then it is unusually badly approximable by algebraic integers of degree at most . By means of Schmidt’s Subspace Theorem we reduce the approximation problem to compute , to an algebraic problem which is trivial if is real but much harder if 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}, 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://archive.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://archive.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://archive.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://archive.numdam.org/item/ASNSP_2009_5_8_2_333_0/
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