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(sq uare-root -1). This is a refinement of a result of Motzkin [2] (see al so [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 a n effective version of a Kronecker approximation theorem; we start wit h an idea of H. Bohr and E. Landau (see e.g. [4]); later we use an est imate of A. Baker for linear forms with logarithms. This and also Rouc he's theorem are then applied in Section 2 to give the result; the req uired irreducibility is guaranteed by the Schonemann-Eisenstein criter ion.