Approximation of complex algebraic numbers by algebraic numbers of bounded degree

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 ∈ C \ R, except for those pairs (n, ξ) such that n is even, n ≥ 6 and n + 3 ≤ deg ξ ≤ 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 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.