Geodesic
Rational Approximations of the Number $\sqrt[3]{3}$
Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 736-747

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We obtain estimates of the form $\|\xi\alpha^n\|>C^k$ for all $k\ge K$ for the values $\xi=3^{1/3}$ and $\alpha=2$, where $K$ is an effective constant and $\|\cdot\|$ denotes the distance to the nearest integer.
Mots-clés : Padé approximant
Keywords: Pfaff–Saalschütz summation formula, Cauchy's theorem, Laplace method, effective estimate, Laplace method, gamma function. 
@article{MZM_2009_86_5_a10,
     author = {Yu. A. Pupyrev},
     title = {Rational {Approximations} of the {Number} $\sqrt[3]{3}$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {736--747},
     publisher = {mathdoc},
     volume = {86},
     number = {5},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a10/}
}
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Yu. A. Pupyrev. Rational Approximations of the Number $\sqrt[3]{3}$. Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 736-747. http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a10/