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Simultaneous rational approximations to certain algebraic numbers
Mathematical Proceedings of the Cambridge Philosophical Society, 1967It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the formis satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such thatThis result would be best possible, for it is well known that (1) has infinitely many solutions ...
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On the Number of Good Simultaneous Approximations to Algebraic Numbers
1991The present work is a continuation of [2] by J. Mueller and the author, where approximations to a single number had been considered. Again it will be convenient to begin with approximations to real numbers in general.
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2019
Summary: Extending his work in Part I, Mahler now shows that the number of representations of a rational integer \(g\) by a binary form \(F(x,y)\) is at most \(O(|g|^{\varepsilon})\), where \(\varepsilon\) is any arbitrarily small positive constant. Reprint of the author's paper [Math. Ann. 108, 37--55 (1933; Zbl 0006.15604; JFM 39.0269.01)].
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Summary: Extending his work in Part I, Mahler now shows that the number of representations of a rational integer \(g\) by a binary form \(F(x,y)\) is at most \(O(|g|^{\varepsilon})\), where \(\varepsilon\) is any arbitrarily small positive constant. Reprint of the author's paper [Math. Ann. 108, 37--55 (1933; Zbl 0006.15604; JFM 39.0269.01)].
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Rational Approximations to Certain Algebraic Numbers
Proceedings of the London Mathematical Society, 1964openaire +1 more source
Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory.
Mathematics of Computation, 1992Eli Passow, Theodore J. Rivlin
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Approximation to real numbers by cubic algebraic integers I
Proceedings of the London Mathematical Society, 2004Damien Roy
exaly
2004
The paper completely solves two questions of K. Mahler and resp. M. Mendes-France, on the rational approximations to the powers of an algebraic numbers (and the continued fractions for these powers). The methods are new and rely on the Subspace Theorem. An Appendix shows that in a sense some of the results are best-possible.
ZANNIER, UMBERTO, U. , CORVAJA, P.
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The paper completely solves two questions of K. Mahler and resp. M. Mendes-France, on the rational approximations to the powers of an algebraic numbers (and the continued fractions for these powers). The methods are new and rely on the Subspace Theorem. An Appendix shows that in a sense some of the results are best-possible.
ZANNIER, UMBERTO, U. , CORVAJA, P.
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