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.
@article{bwmeta1.element.bwnjournal-article-aav63i4p325bwm,
author = {G. J. Rieger},
title = {Effective simultaneous approximation of complex numbers by conjugate algebraic integers},
journal = {Acta Arithmetica},
volume = {64},
year = {1993},
pages = {325-334},
zbl = {0788.11024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav63i4p325bwm}
}
G. J. Rieger. Effective simultaneous approximation of complex numbers by conjugate algebraic integers. Acta Arithmetica, Tome 64 (1993) pp. 325-334. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav63i4p325bwm/
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[002] [3] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, PWN, Warszawa 1974. | Zbl 0276.12002
[003] [4] P. Turán, Nachtrag zu meiner Abhandlung 'On some approximative Dirichlet polynomials in the theory of zeta-function of Riemann', Acta Math. Acad. Sci. Hungar. 10 (1959), 277-298. | Zbl 0103.04503