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Some quantitative results related to Roth's Theorem

Published online by Cambridge University Press:  09 April 2009

E. Bombieri
Affiliation:
School of Mathematics, The Institute for Advanced Study Princeton, New Jersey 08540, U.S.A.
A. J. van der Poorten
Affiliation:
School of Mathematics and Physics, Macquarie UniversityN.S.W. 2109, Australia
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Abstract

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We employ the Dyson's Lemma of Esnault and Viehweg to obtain a new and sharp formulation of Roth's Theorem on the approximation of algebraic numbers by algebraic numbers and apply our arguments to yield a refinement of the Davenport-Roth result on the number of exceptions to Roth's inequality and a sharpening of the Cugiani-Mahler theorem. We improve on the order of magnitude of the results rather than just on the constants involved.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Bombieri, E., ‘On the Thue-Siegel-Dyson theorem’, Acta Math. 148 (1982), 255296.Google Scholar
[2]Bombieri, E. and Vaaler, J., ‘On siegel's lemma’, Invent. Math. 73 (1983), 1132.Google Scholar
[3]Cugiani, Marco, ‘Sull' approssimanilità di un mumero algebrico mediante numeric algebrici di un corpo assengnató’, Boll. Un. Mat. Ital Serie III 14 (1959), 112.Google Scholar
[4]Davenport, H. and Roth, K. F., ‘Rational approximations to algebraic numbers’, Mathe-matika 2 (1955), 160167.Google Scholar
[5]Esnault, H. and Viehweg, E., ‘Dyson's Lemma for polynomials in several variables (and the theorem of Roth)’, Invent. Math. 78 (1984), 445490.Google Scholar
[6]Mahler, K., Lectures on diophantine approximation, Part 1: g-adic numbers and Roth's theorem (University of Notre Damd, 1961).Google Scholar
[7]Mignotte, M., ‘Une généralisation d' un théorème de Cugiani-Mahler’, Acta Arith. 22 (1972), 5767.Google Scholar
[8]Roth, K. F., ‘Rational approximations to algebraic numbers’, Mathematika 2 (1955), 120; Corrigendum, 168.Google Scholar