Effective simultaneous approximation of complex numbers by conjugate algebraic integers

We study effectively the simultaneous approximation of n − 1 different complex numbers by conjugate algebraic integers of degree n over Z( √ −1). This is a refinement of a result of Motzkin [2] (see also [3], p. 50) who has no estimate for the remaining conjugate. If the n−1 different complex numbers lie symmetrically about the real axis, then Z( √ −1) can be replaced by Z. In Section 1 we prove an effective version of a Kronecker approximation theorem; we start with an idea of H. Bohr and E. Landau (see e.g. [4]); later we use an estimate of A. Baker for linear forms with logarithms. This and also Rouché’s theorem are then applied in Section 2 to give the result; the required irreducibility is guaranteed by the Schönemann–Eisenstein criterion.