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Rational approximations of the number $$\root 3 \of {3}$$. (English. Russian original) Zbl 1205.11082
Math. Notes 86, No. 5, 693-703 (2009); translation from Mat. Zametki 86, No. 5, 736-747 (2009).
Apparently, the best known irrationality measure of the number $$3^{1/3}$$ is $$2.69267$$ (due to G. Chudnovsky). This means that the inequality $$|q 3^{1/3}-p|>q^{-1.69267}$$ holds for each sufficiently large positive integer $$q$$ and each integer $$p$$. In this paper the author shows that the constant $$1.69267$$ can be replaced by the constant $$1.50308$$ for a special sequence $$q=2^k$$, $$k \in {\mathbb N}$$, namely, the inequality $$||3^{1/3} 2^k||>0.3568^k$$ holds for each sufficiently large positive integer $$k$$. The proof uses Padé approximants and is effective. The improvement compared to the general estimate comes from the fact the the powers of $$2$$ naturally appear in the denominators $$q_n$$ of corresponding ‘small’ linear forms $$L_n=3^{1/3}-p_n/q_n$$.
##### MSC:
 11J04 Homogeneous approximation to one number
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##### References:
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