Irrationality exponent and rational approximations with prescribed growth

Fischler, Stéphane; Rivoal, Tanguy

doi:10.1090/S0002-9939-09-10084-9

Irrationality exponent and rational approximations with prescribed growth
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by Stéphane Fischler and Tanguy Rivoal PDF

Proc. Amer. Math. Soc. 138 (2010), 799-808 Request permission

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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