Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud[1]

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

Publications mathématiques de Besançon (2014)

  • Issue: 1, page 5-18
  • ISSN: 1958-7236

Abstract

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The Littlewood conjecture in Diophantine approximation claims that inf q 1 q · q α · q β = 0 holds for all real numbers α and β , where · denotes the distance to the nearest integer. Its p -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that inf q 1 q · q α · | q | p = 0 holds for every real number α and every prime number p , where | · | 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.

How to cite

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Bugeaud, Yann. "Around the Littlewood conjecture in Diophantine approximation." Publications mathématiques de Besançon (2014): 5-18. <http://eudml.org/doc/275760>.

@article{Bugeaud2014,
abstract = {The Littlewood conjecture in Diophantine approximation claims that\[ \inf \_\{q \ge 1\} \, q \cdot \Vert q \alpha \Vert \cdot \Vert q \beta \Vert = 0 \]holds for all real numbers $\alpha $ and $\beta $, where $\Vert \cdot \Vert $ denotes the distance to the nearest integer. Its $p$-adic analogue, formulated by de Mathan and Teulié in 2004, asserts that\[ \inf \_\{q \ge 1\} \, q \cdot \Vert q \alpha \Vert \cdot \vert q \vert \_p = 0 \]holds for every real number $\alpha $ 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.},
affiliation = {Mathématiques Université de Strasbourg 7, rue René Descartes, F-67084 Strasbourg France},
author = {Bugeaud, Yann},
journal = {Publications mathématiques de Besançon},
keywords = {Simultaneous approximation; Littlewood conjecture; simultaneous approximation},
language = {eng},
number = {1},
pages = {5-18},
publisher = {Presses universitaires de Franche-Comté},
title = {Around the Littlewood conjecture in Diophantine approximation},
url = {http://eudml.org/doc/275760},
year = {2014},
}

TY - JOUR
AU - Bugeaud, Yann
TI - Around the Littlewood conjecture in Diophantine approximation
JO - Publications mathématiques de Besançon
PY - 2014
PB - Presses universitaires de Franche-Comté
IS - 1
SP - 5
EP - 18
AB - The Littlewood conjecture in Diophantine approximation claims that\[ \inf _{q \ge 1} \, q \cdot \Vert q \alpha \Vert \cdot \Vert q \beta \Vert = 0 \]holds for all real numbers $\alpha $ and $\beta $, where $\Vert \cdot \Vert $ denotes the distance to the nearest integer. Its $p$-adic analogue, formulated by de Mathan and Teulié in 2004, asserts that\[ \inf _{q \ge 1} \, q \cdot \Vert q \alpha \Vert \cdot \vert q \vert _p = 0 \]holds for every real number $\alpha $ 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.
LA - eng
KW - Simultaneous approximation; Littlewood conjecture; simultaneous approximation
UR - http://eudml.org/doc/275760
ER -

References

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  1. B. Adamczewski and Y. Bugeaud, On the Littlewood conjecture in simultaneous Diophantine approximation, J. London Math. Soc. 73 (2006), 355–366. Zbl1093.11052MR2225491
  2. J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge, 2003. Zbl1086.11015MR1997038
  3. D. Badziahin, On multiplicatively badly approximable numbers, Mathematika 59 (2013), 31–55. Zbl1269.11066MR3028170
  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). Zbl1331.11049
  6. D. Badziahin and S. Velani, Multiplicatively badly approximable numbers and the mixed Littlewood conjecture, Adv. Math. 228 (2011), 2766–2796. Zbl1235.11071MR2838058
  7. V. Beresnevich, A. Haynes, and S. Velani, Multiplicative zero-one laws and metric number theory, Acta Arith. 160 (2013), 101–114. Zbl1292.11085MR3105329
  8. M. D. Boshernitzan, Elementary proof of Furstenberg’s Diophantine result, Proc. Amer. Math. Soc. 122 (1994), 67–70. Zbl0815.11036MR1195714
  9. J. Bourgain, E. Lindenstrauss, Ph. Michel, and A. Venkatesh, Some effective results for × a × b , Ergodic Theory Dynam. Systems 29 (2009), 1705–1722. Zbl1237.37009MR2563089
  10. Y. Bugeaud, M. Drmota, and B. de Mathan, On a mixed Littlewood conjecture in Diophantine approximation, Acta Arith. 128 (2007), 107–124. Zbl1209.11064MR2313997
  11. Y. Bugeaud, A. Haynes, and S. Velani, Metric considerations concerning the mixed Littlewood conjecture, Int. J. Number Theory 7 (2011), 593–609. Zbl1259.11069MR2805569
  12. Y. Bugeaud and N. Moshchevitin, Badly approximable numbers and Littlewood-type problems, Math. Proc. Cambridge Phil. Soc. 150 (2011), 215–226. Zbl1231.11071MR2770060
  13. Y. Bugeaud, Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics 193, Cambridge, 2012. Zbl1260.11001MR2953186
  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. Zbl0065.27905MR70653
  15. M. Einsiedler, L. Fishman, and U. Shapira, Diophantine approximation on fractals, Geom. Funct. Anal. 21 (2011), 14–35. Zbl1244.11070MR2773102
  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. Zbl1109.22004MR2247967
  17. M. Einsiedler and D. Kleinbock, Measure rigidity and p -adic Littlewood-type problems, Compositio Math. 143 (2007), 689–702. Zbl1149.11036MR2330443
  18. H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. Zbl0146.28502MR213508
  19. P. Gallagher, Metric simultaneous Diophantine aproximations, J. London Math. Soc. 37 (1962), 387–390. Zbl0124.02902MR157939
  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. Zbl1300.11082MR3118619
  22. A. Haynes, J. L. Jensen, and S. Kristensen, Metrical musings on Littlewood and friends, Proc. Amer. Math. Soc. 142 (2014), 457–466. Zbl1327.11050MR3133988
  23. A. Haynes and S. Munday, Diophantine approximation and coloring, Amer. Math. Monthly. To appear. Zbl1328.05065
  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. Zbl1235.37004MR2827904
  25. B. de Mathan, Conjecture de Littlewood et récurrences linéaires, J. Théor. Nombres Bordeaux 13 (2003), 249–266. Zbl1045.11048
  26. B. de Mathan et O. Teulié, Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245. Zbl1162.11361
  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. Zbl0019.33502MR1507944
  29. M. Morse and G. A. Hedlund, Symbolic dynamics II, Amer. J. Math. 62 (1940), 1–42. Zbl0022.34003MR745
  30. L. G. Peck, Simultaneous rational approximations to algebraic numbers, Bull. Amer. Math. Soc. 67 (1961), 197–201. Zbl0098.26302MR122772
  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. Zbl1215.05074MR2601556
  32. A. D. Pollington and S. Velani, On a problem in simultaneous Diophantine approximation: Littlewood’s conjecture, Acta Math. 185 (2000), 287–306. Zbl0970.11026MR1819996
  33. U. Shapira, A solution to a problem of Cassels and Diophantine properties of cubic numbers, Ann. of Math. 173 (2011), 543–557. Zbl1242.11046MR2753608
  34. D. C. Spencer, The lattice points of tetrahedra, J. Math. Phys. Mass. Inst. Tech. 21 (1942), 189–197. Zbl0060.11501MR7767
  35. A. Venkatesh, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 117–134. Zbl1194.11075MR2358379

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