Geodesic
Rational Approximations of the Number $\sqrt[3]{3}$
Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 736-747 Cet article a éte moissonné depuis la source Math-Net.Ru

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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, Pfaff–Saalschütz summation formula
Keywords: Cauchy's theorem, Laplace method, effective estimate, Laplace method, gamma function. 
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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/

[1] K. Mahler, “On the fractional parts of powers of real numbers. II”, Mathematika, 4 (1957), 122–124 | MR | Zbl

[2] A. Baker, J. Coates, “Fractional parts of powers of rationals”, Math. Proc. Cambridge Philos. Soc., 77 (1975), 269–279 | DOI | MR | Zbl

[3] F. Beukers, “Fractional parts of powers of rationals”, Math. Proc. Cambridge Philos. Soc., 90:1 (1981), 13–20 | DOI | MR | Zbl

[4] A. K. Dubitskas, “Otsenka snizu velichiny $\|(3/2)^k\|$”, UMN, 45:4 (1990), 153–154 | MR | Zbl

[5] L. Habsieger, “Explicit lower bounds for $\|(3/2)^k\|$”, Acta Arith., 106:3 (2003), 299–309 | DOI | MR | Zbl

[6] W. Zudilin, “A new lower bound for $\|(3/2)^k\|$”, J. Théor. Nombres Bordeaux, 19:1 (2007), 311–323 | MR | Zbl

[7] Yu. A. Pupyrev, “Effektivizatsiya nizhnei otsenki dlya $\|(4/3)^k\|$”, Matem. zametki, 85:6 (2009), 927–935

[8] G. V. Chudnovsky, “On the method of Thue–Siegel”, Ann. of Math. (2), 117:2 (1983), 325–382 | DOI | MR | Zbl

[9] L. J. Slater, Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, 1966 | MR | Zbl