Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T14:27:31.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Some quantitative results related to Roth's Theorem

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  09 April 2009

E. Bombieri  and
A. J. van der Poorten
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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Secondary: 11J68: Approximation to algebraic numbers
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 2 , October 1988 , pp. 233 - 248
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Bombieri, E., ‘On the Thue-Siegel-Dyson theorem’, Acta Math. 148 (1982), 255–296.CrossRefGoogle Scholar
[2]Bombieri, E. and Vaaler, J., ‘On siegel's lemma’, Invent. Math. 73 (1983), 11–32.CrossRefGoogle 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), 1–12.Google Scholar
[4]Davenport, H. and Roth, K. F., ‘Rational approximations to algebraic numbers’, Mathe-matika 2 (1955), 160–167.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), 445–490.CrossRefGoogle 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), 57–67.CrossRefGoogle Scholar
[8]Roth, K. F., ‘Rational approximations to algebraic numbers’, Mathematika 2 (1955), 1–20; Corrigendum, 168.CrossRefGoogle Scholar