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The number of exceptional approximations in Roth's theorem

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  09 April 2009

Wolfgang M. Schmidt
Wolfgang M. Schmidt
Affiliation:
Department of MathematicsUniversity of Colorado at BoulderBoulder, Colorado, USA
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Abstract

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Roth's Theorem says that given ρ < 2 and an algebraic number α, all but finitely many rational numbers x/y satisfy |α - (x/y)|< |y|-ρ. We give upper bounds for the number of these exceptional rationals when 3 ≤ ρ ≤ d, where d is the degree of α. Our result suplements bounds given by Bombieri and Van der Poorten when 2 > ρ ≤ 3; naturally the bounds become smaller as ρ increases.

MSC classification

Secondary: 11J68: Approximation to algebraic numbers
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 59 , Issue 3 , December 1995 , pp. 375 - 383
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Bombieri, E. and Van der Poorten, A. J., ‘Some quantitative results related to Roth's theorem’, J. Austral. Math. Soc. (Series A) 45 (1988), 233–248.CrossRefGoogle Scholar
[2]Schmidt, W. M., Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics 1467 (Springer, Berlin, 1991).CrossRefGoogle Scholar
[3]Silverman, Joseph H., The arithmetic of elliptic curves, Graduate Texts in Mathematics (Springer, Berlin, 1986).CrossRefGoogle Scholar