Irrationality exponent and rational approximations with prescribed growth

Details Irrationality exponent and rational approximations with prescribed growth Irrationality exponent and rational approximations with prescribed growth

2009-10-23

arxiv.org/abs/0910.4448v1

Proc. Amer. Math. Soc. 138.8 (2010), 799-808

Mathematics

Number Theory

11 pages, to appear in Proc. Amer. Math. Soc

Scientific paper

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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