Comptes Rendus
Théorie des nombres
Fractional parts of powers of real algebraic numbers
Yann Bugeaud1, 2
1 Institut universitaire de France
2 Université de Strasbourg, Mathématiques, 7 rue René Descartes, 67084 Strasbourg, France
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 459-466.

Let α be a real algebraic number greater than 1. We establish an effective lower bound for the distance between an integral power of α and its nearest integer.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.314
Classification : 11J68, 11J86, 11R06
Mots clés : Approximation to algebraic numbers, Linear forms in logarithms, Pisot number
Yann Bugeaud 1, 2

1 Institut universitaire de France
2 Université de Strasbourg, Mathématiques, 7 rue René Descartes, 67084 Strasbourg, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2022__360_G5_459_0,
     author = {Yann Bugeaud},
     title = {Fractional parts of powers of real algebraic numbers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {459--466},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.314},
     language = {en},
}
TY  - JOUR
AU  - Yann Bugeaud
TI  - Fractional parts of powers of real algebraic numbers
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 459
EP  - 466
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.314
LA  - en
ID  - CRMATH_2022__360_G5_459_0
ER  - 
%0 Journal Article
%A Yann Bugeaud
%T Fractional parts of powers of real algebraic numbers
%J Comptes Rendus. Mathématique
%D 2022
%P 459-466
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.314
%G en
%F CRMATH_2022__360_G5_459_0
Yann Bugeaud. Fractional parts of powers of real algebraic numbers. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 459-466. doi : 10.5802/crmath.314. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.314/

[1] Alan Baker; John H. Coates Fractional parts of powers of rationals, Math. Proc. Camb. Philos. Soc., Volume 77 (1975), pp. 269-279 | DOI | MR | Zbl

[2] Mark Bauer; Michael A. Bennett Applications of the hypergeometric method to the generalized Ramanujan–Nagell equation, Ramanujan J., Volume 6 (2002) no. 2, pp. 209-270 | DOI | MR | Zbl

[3] Michael A. Bennett; Yann Bugeaud Effective results for restricted rational approximation to quadratic irrationals, Acta Arith., Volume 155 (2012) no. 3, pp. 259-269 | DOI | MR | Zbl

[4] Frits Beukers Fractional parts of powers of rationals, Math. Proc. Camb. Philos. Soc., Volume 90 (1981), pp. 13-20 | DOI | MR | Zbl

[5] Frits Beukers On the generalized Ramanujan-Nagell equation I, Acta Arith., Volume 38 (1981), pp. 389-410 | DOI | MR | Zbl

[6] David W. Boyd Irreducible polynomials with many roots of maximal modulus, Acta Arith., Volume 68 (1994) no. 1, pp. 85-88 | DOI | MR | Zbl

[7] Yann Bugeaud Linear forms in two m-adic logarithms and applications to Diophantine problems, Compos. Math., Volume 132 (2002) no. 2, pp. 137-158 | DOI | MR | Zbl

[8] Yann Bugeaud Linear forms in logarithms and applications, IRMA Lectures in Mathematics and Theoretical Physics, 28, European Mathematical Society, 2018 | DOI | Zbl

[9] Yann Bugeaud Effective simultaneous rational approximation to pairs of real quadratic numbers, Mosc. J. Comb. Number Theory, Volume 9 (2020) no. 4, pp. 353-360 | DOI | MR | Zbl

[10] Yann Bugeaud; Artūras Dubickas On a problem of Mahler and Szekeres on approximation by roots of integers, Mich. Math. J., Volume 56 (2008) no. 3, pp. 703-715 | MR | Zbl

[11] Pietro Corvaja; Umberto Zannier On the rational approximations to the powers of an algebraic number: solution of two problems of Mahler and Mendès France, Acta Math., Volume 193 (2004) no. 2, pp. 175-191 | DOI

[12] Artūras Dubickas Roots of unity as quotients of two conjugate algebraic numbers, Glas. Mat., III. Ser., Volume 52 (2017) no. 2, pp. 235-240 | DOI | MR | Zbl

[13] Jan-Hendrik Evertse; Kálmán Győry Unit equations in Diophantine number theory, Cambridge Studies in Advanced Mathematics, 146, Cambridge University Press, 2015 | DOI

[14] Naum I. Fel’dman Improved estimate for a linear form of the logarithms of algebraic numbers, Mat. Sb., Volume 77 (1968), pp. 256-270 English translation in Math. USSR. Sb. 6 (1968), p. 393–406

[15] Kurt Mahler On the fractional parts of the powers of a rational number, II, Mathematika, Volume 4 (1957), pp. 122-124 | DOI | MR | Zbl

[16] Kurt Mahler; George Szekeres On the approximation of real numbers by roots of integers, Acta Arith., Volume 12 (1967), pp. 315-320 | DOI | MR | Zbl

[17] Michel Waldschmidt Diophantine Approximation on Linear Algebraic Groups. Transcendence Properties of the Exponential Function in Several Variables, Grundlehren der Mathematischen Wissenschaften, 326, Springer, 2000 | DOI

[18] Kunrui Yu p–adic logarithmic forms and group varieties, III, Forum Math., Volume 19 (2007) no. 2, pp. 187-280 | MR | Zbl

[19] Wadim Zudilin A new lower bound for ∥(3/2) k ∥, J. Théor. Nombres Bordeaux, Volume 19 (2007) no. 1, pp. 311-323 | DOI | Numdam | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Fractional parts of powers and Sturmian words

Yann Bugeaud; Artūras Dubickas

C. R. Math (2005)

Sur la complexité des nombres algébriques

Boris Adamczewski; Yann Bugeaud; Florian Luca

C. R. Math (2004)