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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....

Effective solution of the D(-1)-quadruple conjecture

Andrej Dujella, Alan Filipin, Clemens Fuchs (2007)

Acta Arithmetica

Equations diophantiennes exponentielles

Michel LAURENT (1983/1984)

Seminaire de Théorie des Nombres de Bordeaux

Exceptional units and cyclic resultants

C. L. Stewart (2012)

Acta Arithmetica

Exceptional units and numbers of small Mahler measure.

Silverman, Joseph H. (1995)

Experimental Mathematics

Extensions of the Cugiani-Mahler theorem

Yann Bugeaud (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In 1955, Roth established that if $ξ$ is an irrational number such that there are a positive real number $ε$ and infinitely many rational numbers $p / q$ with $q \geq 1$ and $| ξ - p / q | < q^{- 2 - ε}$ , then $ξ$ is transcendental. A few years later, Cugiani obtained the same conclusion with $ε$ replaced by a function $q \mapsto ε (q)$ that decreases very slowly to zero, provided that the sequence of rational solutions to $| ξ - p / q | < q^{- 2 - ε (q)}$ is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it to simultaneous...

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