The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture
Author:
Akshay Venkatesh
Journal:
Bull. Amer. Math. Soc. 45 (2008), 117-134
MSC (2000):
Primary 11J13, 37A35, 33A45, 11H46
DOI:
https://doi.org/10.1090/S0273-0979-07-01194-9
Published electronically:
October 29, 2007
MathSciNet review:
2358379
Full-text PDF Free Access
References | Similar Articles | Additional Information
- 1. S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 91–137. MR 1237827
- 2.
J. Ellenberg and A. Venkatesh.
Local-global principles for representations of quadratic forms.
arxiv: math.NT/0604232. - 3. Manfred Einsiedler and Anatole Katok, Rigidity of measures—the high entropy case and non-commuting foliations, Israel J. Math. 148 (2005), 169–238. Probability in mathematics. MR 2191228, https://doi.org/10.1007/BF02775436
- 4. Manfred Einsiedler and Anatole Katok, Invariant measures on 𝐺/Γ for split simple Lie groups 𝐺, Comm. Pure Appl. Math. 56 (2003), no. 8, 1184–1221. Dedicated to the memory of Jürgen K. Moser. MR 1989231, https://doi.org/10.1002/cpa.10092
- 5. Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss, Invariant measures and the set of exceptions to Littlewood’s conjecture, Ann. of Math. (2) 164 (2006), no. 2, 513–560. MR 2247967, https://doi.org/10.4007/annals.2006.164.513
- 6. Manfred Einsiedler and Elon Lindenstrauss, Diagonalizable flows on locally homogeneous spaces and number theory, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1731–1759. MR 2275667
- 7.
Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel and Akshay Venkatesh.
The distribution of periodic torus orbits on homogeneous spaces.
arxiv: math.DS/0607815. - 8. Boris Kalinin and Ralf Spatzier, Rigidity of the measurable structure for algebraic actions of higher-rank Abelian groups, Ergodic Theory Dynam. Systems 25 (2005), no. 1, 175–200. MR 2122918, https://doi.org/10.1017/S014338570400046X
- 9. A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems 16 (1996), no. 4, 751–778. MR 1406432, https://doi.org/10.1017/S0143385700009081
- 10. A. Katok and R. J. Spatzier, Corrections to: “Invariant measures for higher-rank hyperbolic abelian actions” [Ergodic Theory Dynam. Systems 16 (1996), no. 4, 751–778; MR1406432 (97d:58116)], Ergodic Theory Dynam. Systems 18 (1998), no. 2, 503–507. MR 1619571, https://doi.org/10.1017/S0143385798110969
- 11. Y. Katznelson, Chromatic numbers of Cayley graphs on ℤ and recurrence, Combinatorica 21 (2001), no. 2, 211–219. Paul Erdős and his mathematics (Budapest, 1999). MR 1832446, https://doi.org/10.1007/s004930100019
- 12. F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539. MR 819556, https://doi.org/10.2307/1971328
- 13. F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539. MR 819556, https://doi.org/10.2307/1971328
- 14.
Elon Lindenstrauss.
Arithmetic quantum unique ergodicity and adelic dynamics.
Proceedings of Current Developments in Mathematics conference (2004), to appear. - 15. Elon Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2) 163 (2006), no. 1, 165–219. MR 2195133, https://doi.org/10.4007/annals.2006.163.165
- 16. Elon Lindenstrauss, Rigidity of multiparameter actions, Israel J. Math. 149 (2005), 199–226. Probability in mathematics. MR 2191215, https://doi.org/10.1007/BF02772541
- 17. Gregory Margulis, Problems and conjectures in rigidity theory, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 161–174. MR 1754775
- 18. G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. 116 (1994), no. 1-3, 347–392. MR 1253197, https://doi.org/10.1007/BF01231565
- 19. Andrew D. Pollington and Sanju L. Velani, On a problem in simultaneous Diophantine approximation: Littlewood’s conjecture, Acta Math. 185 (2000), no. 2, 287–306. MR 1819996, https://doi.org/10.1007/BF02392812
- 20. Marina Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2) 118 (1983), no. 2, 277–313. MR 717825, https://doi.org/10.2307/2007030
- 21. Marina Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), no. 3, 545–607. MR 1135878, https://doi.org/10.2307/2944357
- 22. Marina Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235–280. MR 1106945, https://doi.org/10.1215/S0012-7094-91-06311-8
- 23. Carl Ludwig Siegel, Lectures on the geometry of numbers, Springer-Verlag, Berlin, 1989. Notes by B. Friedman; Rewritten by Komaravolu Chandrasekharan with the assistance of Rudolf Suter; With a preface by Chandrasekharan. MR 1020761
- 24.
Lior Silberman and Akshay Venkatesh.
On quantum unique ergodicity for locally symmetric spaces.
math.RT/0407413, to appear, GAFA, 17 (3) (2007), 960-998. - 25. Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
- 26.
David Witte.
Ratner's theorems on unipotent flows.
Chicago Lectures in Mathematics Series, University of Chicago Press, Chicago, IL, 2005.
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Additional Information
Akshay Venkatesh
Affiliation:
Department of Mathematics, Courant Institute, New York University, New York, New York 10012
DOI:
https://doi.org/10.1090/S0273-0979-07-01194-9
Received by editor(s):
May 11, 2007
Received by editor(s) in revised form:
May 28, 2007
Published electronically:
October 29, 2007
Additional Notes:
This article is based on a lecture presented January 7, 2007, as part of the Current Events Bulletin at the Joint Mathematics Meetings in New Orleans, LA
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.