Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud

doi:10.5802/pmb.1

Yann Bugeaud ¹

¹ Mathématiques Université de Strasbourg 7, rue René Descartes, F-67084 Strasbourg France

Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 5-18.

Abstract
Résumé

The Littlewood conjecture in Diophantine approximation claims that

inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot ∥ q β ∥ = 0

holds for all real numbers $α$ and $β$ , where $∥ \cdot ∥$ denotes the distance to the nearest integer. Its $p$ -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that

inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot {| q |}_{p} = 0

holds for every real number $α$ and every prime number $p$ , where ${| \cdot |}_{p}$ denotes the $p$ -adic absolute value normalized by ${| p |}_{p} = p^{- 1}$ . We survey the known results on these conjectures and highlight recent developments.

En approximation diophantienne, la conjecture de Littlewood stipule que tous les nombres réels $α$ et $β$ vérifient

inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot ∥ q β ∥ = 0,

où $∥ \cdot ∥$ désigne la distance à l’entier le plus proche. Son analogue $p$ -adique, formulé par de Mathan et Teulié en 2004, affirme que l’égalité

inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot {| q |}_{p} = 0

est valable pour tout nombre réel $α$ et tout nombre premier $p$ , où ${| \cdot |}_{p}$ est la valeur absolue $p$ -adique normalisée par ${| p |}_{p} = p^{- 1}$ . Nous donnons un survol des résultats connus sur ces conjectures en insistant sur les développements récents.

Received: 2014-05-24
Published online: 2015-04-13

Zbl: 1367.11059

DOI: 10.5802/pmb.1

Classification: 11J04, 11J13, 11J61
Keywords: Simultaneous approximation, Littlewood conjecture

Author's affiliations:

Yann Bugeaud ¹

¹ Mathématiques Université de Strasbourg 7, rue René Descartes, F-67084 Strasbourg France

@article{PMB_2014___1_5_0,
     author = {Yann Bugeaud},
     title = {Around the {Littlewood} conjecture in {Diophantine} approximation},
     journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
     pages = {5--18},
     publisher = {Presses universitaires de Franche-Comt\'e},
     number = {1},
     year = {2014},
     doi = {10.5802/pmb.1},
     zbl = {1367.11059},
     language = {en},
     url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.1/}
}

TY  - JOUR
AU  - Yann Bugeaud
TI  - Around the Littlewood conjecture in Diophantine approximation
JO  - Publications mathématiques de Besançon. Algèbre et théorie des nombres
PY  - 2014
SP  - 5
EP  - 18
IS  - 1
PB  - Presses universitaires de Franche-Comté
UR  - https://pmb.centre-mersenne.org/articles/10.5802/pmb.1/
DO  - 10.5802/pmb.1
LA  - en
ID  - PMB_2014___1_5_0
ER  -

%0 Journal Article
%A Yann Bugeaud
%T Around the Littlewood conjecture in Diophantine approximation
%J Publications mathématiques de Besançon. Algèbre et théorie des nombres
%D 2014
%P 5-18
%N 1
%I Presses universitaires de Franche-Comté
%U https://pmb.centre-mersenne.org/articles/10.5802/pmb.1/
%R 10.5802/pmb.1
%G en
%F PMB_2014___1_5_0

Yann Bugeaud. Around the Littlewood conjecture in Diophantine approximation. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 5-18. doi : 10.5802/pmb.1. https://pmb.centre-mersenne.org/articles/10.5802/pmb.1/

References
Cited by

[1] B. Adamczewski and Y. Bugeaud, On the Littlewood conjecture in simultaneous Diophantine approximation, J. London Math. Soc. 73 (2006), 355–366. | DOI | MR | Zbl

[2] J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge, 2003. | DOI | Zbl

[3] D. Badziahin, On multiplicatively badly approximable numbers, Mathematika 59 (2013), 31–55. | DOI | MR | Zbl

[4] D. Badziahin, On the continued fraction expansion of potential counterexamples to the $p$ -adic Littlewood conjecture, preprint (arXiv:1406.3594).

[5] D. Badziahin, Y. Bugeaud, M. Einsiedler and D. Kleinbock, On the complexity of a putative counterexample to the $p$ -adic Littlewood conjecture, preprint (arXiv:1405.5545). | DOI | MR | Zbl

[6] D. Badziahin and S. Velani, Multiplicatively badly approximable numbers and the mixed Littlewood conjecture, Adv. Math. 228 (2011), 2766–2796. | DOI | MR | Zbl

[7] V. Beresnevich, A. Haynes, and S. Velani, Multiplicative zero-one laws and metric number theory, Acta Arith. 160 (2013), 101–114. | DOI | MR | Zbl

[8] M. D. Boshernitzan, Elementary proof of Furstenberg’s Diophantine result, Proc. Amer. Math. Soc. 122 (1994), 67–70. | DOI | MR

[9] J. Bourgain, E. Lindenstrauss, Ph. Michel, and A. Venkatesh, Some effective results for $\times a \times b$ , Ergodic Theory Dynam. Systems 29 (2009), 1705–1722. | DOI | MR | Zbl

[10] Y. Bugeaud, M. Drmota, and B. de Mathan, On a mixed Littlewood conjecture in Diophantine approximation, Acta Arith. 128 (2007), 107–124. | DOI | MR | Zbl

[11] Y. Bugeaud, A. Haynes, and S. Velani, Metric considerations concerning the mixed Littlewood conjecture, Int. J. Number Theory 7 (2011), 593–609. | DOI | MR | Zbl

[12] Y. Bugeaud and N. Moshchevitin, Badly approximable numbers and Littlewood-type problems, Math. Proc. Cambridge Phil. Soc. 150 (2011), 215–226. | DOI | MR | Zbl

[13] Y. Bugeaud, Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics 193, Cambridge, 2012. | DOI | Zbl

[14] J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms and indefinite ternary quadratic forms, Philos. Trans. Roy. Soc. London, Ser. A, 248 (1955), 73–96. | DOI | MR | Zbl

[15] M. Einsiedler, L. Fishman, and U. Shapira, Diophantine approximation on fractals, Geom. Funct. Anal. 21 (2011), 14–35. | DOI | MR | Zbl

[16] M. Einsiedler, A. Katok, and E. Lindenstrauss, Invariant measures and the set of exceptions to the Littlewood conjecture, Ann. of Math. 164 (2006), 513–560. | DOI | MR | Zbl

[17] M. Einsiedler and D. Kleinbock, Measure rigidity and $p$ -adic Littlewood-type problems, Compositio Math. 143 (2007), 689–702. | DOI | MR | Zbl

[18] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. | DOI | MR | Zbl

[19] P. Gallagher, Metric simultaneous Diophantine aproximations, J. London Math. Soc. 37 (1962), 387–390. | DOI | MR | Zbl

[20] A. Gorodnik and P. Vishe, Inhomogeneous multiplicative Littlewood conjecture and logarithmic savings. In preparation.

[21] S. Harrap and A. Haynes, The mixed Littlewood conjecture for pseudo-absolute values, Math. Ann. 357 (2013), 941–960. | DOI | MR | Zbl

[22] A. Haynes, J. L. Jensen, and S. Kristensen, Metrical musings on Littlewood and friends, Proc. Amer. Math. Soc. 142 (2014), 457–466. | DOI | MR | Zbl

[23] A. Haynes and S. Munday, Diophantine approximation and coloring, Amer. Math. Monthly. To appear. | DOI | MR | Zbl

[24] E. Lindenstrauss, Equidistribution in homogeneous spaces and number theory. In: Proceedings of the International Congress of Mathematicians. Volume I, 531–557, Hindustan Book Agency, New Delhi, 2010. | DOI | Zbl

[25] B. de Mathan, Conjecture de Littlewood et récurrences linéaires, J. Théor. Nombres Bordeaux 13 (2003), 249–266. | DOI | Zbl

[26] B. de Mathan et O. Teulié, Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245. | DOI | Zbl

[27] H. L. Montgomery, Littlewood’s work in number theory, Bull. London Math. Soc. 11 (1979), 78–86.

[28] M. Morse and G. A. Hedlund, Symbolic dynamics, Amer. J. Math. 60 (1938), 815–866. | DOI | MR | Zbl

[29] M. Morse and G. A. Hedlund, Symbolic dynamics II, Amer. J. Math. 62 (1940), 1–42. | DOI | MR | Zbl

[30] L. G. Peck, Simultaneous rational approximations to algebraic numbers, Bull. Amer. Math. Soc. 67 (1961), 197–201. | DOI | MR | Zbl

[31] Yu. Peres and W. Schlag, Two Erdős problems on lacunary sequences: chromatic numbers and Diophantine approximations, Bull. Lond. Math. Soc. 42 (2010), 295–300. | DOI | Zbl

[32] A. D. Pollington and S. Velani, On a problem in simultaneous Diophantine approximation: Littlewood’s conjecture, Acta Math. 185 (2000), 287–306. | DOI | MR | Zbl

[33] U. Shapira, A solution to a problem of Cassels and Diophantine properties of cubic numbers, Ann. of Math. 173 (2011), 543–557. | DOI | MR | Zbl

[34] D. C. Spencer, The lattice points of tetrahedra, J. Math. Phys. Mass. Inst. Tech. 21 (1942), 189–197. | DOI | MR | Zbl

[35] A. Venkatesh, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 117–134. | DOI | MR | Zbl

Cited by Sources:

Publications Mathématiques de Besançon

Algèbre et Théorie des Nombres