Publications Mathématiques de Besançon
Algèbre et Théorie des Nombres
Avec cedram.org english version
Table des matières de ce fascicule | Article suivant
Yann Bugeaud
Around the Littlewood conjecture in Diophantine approximation
Publications mathématiques de Besançon no. 1 (2014), p. 5-18, doi: 10.5802/pmb.1
Article PDF
Class. Math.: 11J04, 11J13, 11J61
Mots clés: Simultaneous approximation, Littlewood conjecture

Résumé

En approximation diophantienne, la conjecture de Littlewood stipule que tous les nombres réels $\alpha $ et $\beta $ vérifient

$$ \inf _{q \ge 1} \, q \cdot \Vert q \alpha \Vert \cdot \Vert q \beta \Vert = 0, $$

où $\Vert \cdot \Vert $ 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

$$ \inf _{q \ge 1} \, q \cdot \Vert q \alpha \Vert \cdot \vert q \vert _p = 0 $$

est valable pour tout nombre réel $\alpha $ 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.

Bibliographie

[1] B. Adamczewski and Y. Bugeaud, On the Littlewood conjecture in simultaneous Diophantine approximation, J. London Math. Soc. 73 (2006), 355–366.  MR 2225491 |  Zbl 1093.11052
[2] J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge, 2003.  MR 1997038 |  Zbl 1086.11015
[3] D. Badziahin, On multiplicatively badly approximable numbers, Mathematika 59 (2013), 31–55.  MR 3028170 |  Zbl 1269.11066
[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).
[6] D. Badziahin and S. Velani, Multiplicatively badly approximable numbers and the mixed Littlewood conjecture, Adv. Math. 228 (2011), 2766–2796.  MR 2838058 |  Zbl 1235.11071
[7] V. Beresnevich, A. Haynes, and S. Velani, Multiplicative zero-one laws and metric number theory, Acta Arith. 160 (2013), 101–114.  MR 3105329 |  Zbl 1292.11085
[8] M. D. Boshernitzan, Elementary proof of Furstenberg’s Diophantine result, Proc. Amer. Math. Soc. 122 (1994), 67–70.  MR 1195714 |  Zbl 0815.11036
[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.  MR 2563089 |  Zbl 1237.37009
[10] Y. Bugeaud, M. Drmota, and B. de Mathan, On a mixed Littlewood conjecture in Diophantine approximation, Acta Arith. 128 (2007), 107–124.  MR 2313997 |  Zbl 1209.11064
[11] Y. Bugeaud, A. Haynes, and S. Velani, Metric considerations concerning the mixed Littlewood conjecture, Int. J. Number Theory 7 (2011), 593–609.  MR 2805569 |  Zbl 1259.11069
[12] Y. Bugeaud and N. Moshchevitin, Badly approximable numbers and Littlewood-type problems, Math. Proc. Cambridge Phil. Soc. 150 (2011), 215–226.  MR 2770060 |  Zbl 1231.11071
[13] Y. Bugeaud, Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics 193, Cambridge, 2012.  MR 2953186 |  Zbl 1260.11001
[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.  MR 70653 |  Zbl 0065.27905
[15] M. Einsiedler, L. Fishman, and U. Shapira, Diophantine approximation on fractals, Geom. Funct. Anal. 21 (2011), 14–35.  MR 2773102 |  Zbl 1244.11070
[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.  MR 2247967 |  Zbl 1109.22004
[17] M. Einsiedler and D. Kleinbock, Measure rigidity and $p$-adic Littlewood-type problems, Compositio Math. 143 (2007), 689–702.  MR 2330443 |  Zbl 1149.11036
[18] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49.  MR 213508 |  Zbl 0146.28502
[19] P. Gallagher, Metric simultaneous Diophantine aproximations, J. London Math. Soc. 37 (1962), 387–390.  MR 157939 |  Zbl 0124.02902
[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.  MR 3118619 |  Zbl 1300.11082
[22] A. Haynes, J. L. Jensen, and S. Kristensen, Metrical musings on Littlewood and friends, Proc. Amer. Math. Soc. 142 (2014), 457–466.  MR 3133988
[23] A. Haynes and S. Munday, Diophantine approximation and coloring, Amer. Math. Monthly. To appear.
[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.  MR 2827904 |  Zbl 1235.37004
[25] B. de Mathan, Conjecture de Littlewood et récurrences linéaires, J. Théor. Nombres Bordeaux 13 (2003), 249–266. Cedram |  Zbl 1045.11048
[26] B. de Mathan et O. Teulié, Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245.  Zbl 1162.11361
[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.  MR 1507944 |  JFM 64.0798.04
[29] M. Morse and G. A. Hedlund, Symbolic dynamics II, Amer. J. Math. 62 (1940), 1–42.  MR 745 |  Zbl 0022.34003 |  JFM 66.0188.03
[30] L. G. Peck, Simultaneous rational approximations to algebraic numbers, Bull. Amer. Math. Soc. 67 (1961), 197–201.  MR 122772 |  Zbl 0098.26302
[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.  MR 2601556 |  Zbl 1215.05074
[32] A. D. Pollington and S. Velani, On a problem in simultaneous Diophantine approximation: Littlewood’s conjecture, Acta Math. 185 (2000), 287–306.  MR 1819996 |  Zbl 0970.11026
[33] U. Shapira, A solution to a problem of Cassels and Diophantine properties of cubic numbers, Ann. of Math. 173 (2011), 543–557.  MR 2753608 |  Zbl 1242.11046
[34] D. C. Spencer, The lattice points of tetrahedra, J. Math. Phys. Mass. Inst. Tech. 21 (1942), 189–197.  MR 7767 |  Zbl 0060.11501
[35] A. Venkatesh, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 117–134.  MR 2358379 |  Zbl 1194.11075

UFC LMB PUFC