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(square-root -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(square-root -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 Rouche's theorem are then applied in Section 2 to give the result; the required irreducibility is guaranteed by the Schonemann-Eisenstein criterion.