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Irrationality exponent and rational approximations with prescribed growth

Authors: Stéphane Fischler and Tanguy Rivoal
Journal: Proc. Amer. Math. Soc. 138 (2010), 799-808
MSC (2000): Primary 11J82; Secondary 11J04, 11J13, 11J72
Published electronically: October 20, 2009
MathSciNet review: 2566545
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Abstract: Let $ \xi$ be a real irrational number. We are interested in sequences of linear forms in 1 and $ \xi$, with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of $ \xi$. We prove that this condition is actually sufficient, even for arbitrary rates of growth and decrease. We also make some remarks and ask some questions about multivariate generalizations connected to Fischler-Zudilin's new proof of Nesterenko's linear independence criterion.

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Additional Information

Stéphane Fischler
Affiliation: Université Paris-Sud, Laboratoire de Mathématiques d’Orsay, Orsay cedex, F-91405, France – and – CNRS, Orsay cedex, F-91405, France

Tanguy Rivoal
Affiliation: Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères cedex, France

Received by editor(s): June 9, 2009
Published electronically: October 20, 2009
Communicated by: Ken Ono
Article copyright: © Copyright 2009 American Mathematical Society