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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - Yu. A. Pupyrev
TI  - Rational Approximations of the Number $\sqrt[3]{3}$
JO  - Matematičeskie zametki
PY  - 2009
SP  - 736
EP  - 747
VL  - 86
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a10/
LA  - ru
ID  - MZM_2009_86_5_a10
ER  - 
%0 Journal Article
%A Yu. A. Pupyrev
%T Rational Approximations of the Number $\sqrt[3]{3}$
%J Matematičeskie zametki
%D 2009
%P 736-747
%V 86
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a10/
%G ru
%F MZM_2009_86_5_a10
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