J Austral Math Soc Ser A 59 pp375--383, 1995.

The Number of Exceptional Approximations in Roth's Theorem

Wolfgang M. Schmidt

Abstract

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

1991 AMS Subject Classification: 11J68

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Authors

Wolfgang M. Schmidt

Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado, USA.