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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
Posted:
October 29, 2007
MathSciNet review:
2358379
Full-text PDF
References |
Similar Articles |
Additional Information
- 1.
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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
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Littlewood’s conjecture, Ann. of Math. (2) 164
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The distribution of periodic torus orbits on homogeneous spaces. arxiv: math.DS/0607815.
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- 9.
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actions, Ergodic Theory Dynam. Systems 16 (1996),
no. 4, 751–778. MR 1406432
(97d:58116), http://dx.doi.org/10.1017/S0143385700009081
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Katok and R.
J. Spatzier, Corrections to: “Invariant measures for
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Systems 16 (1996), no. 4, 751–778; MR1406432 (97d:58116)],
Ergodic Theory Dynam. Systems 18 (1998), no. 2,
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Young, The metric entropy of diffeomorphisms. II. Relations between
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122 (1985), no. 3, 540–574. MR 819557
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Lindenstrauss, Invariant measures and arithmetic quantum unique
ergodicity, Ann. of Math. (2) 163 (2006), no. 1,
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Lindenstrauss, Rigidity of multiparameter actions, Israel J.
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(2001d:22008)
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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
(95k:22013), http://dx.doi.org/10.1007/BF01231565
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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
(2002a:11076), http://dx.doi.org/10.1007/BF02392812
- 20.
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Ratner, On Raghunathan’s measure conjecture, Ann. of
Math. (2) 134 (1991), no. 3, 545–607. MR 1135878
(93a:22009), http://dx.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
(93f:22012), http://dx.doi.org/10.1215/S0012-7094-91-06311-8
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(91d:11070)
- 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
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(84e:28017)
- 26.
David Witte.
Ratner's theorems on unipotent flows. Chicago Lectures in Mathematics Series, University of Chicago Press, Chicago, IL, 2005.
- 1.
- S. G. Dani and G. A. Margulis.
Limit distributions of orbits of unipotent flows and values of quadratic forms. In I. M. Gelfand Seminar, volume 16 of Adv. Soviet Math., pages 91-137. Amer. Math. Soc., Providence, RI, 1993. MR 1237827 (95b:22024)
- 2.
- J. Ellenberg and A. Venkatesh.
Local-global principles for representations of quadratic forms. arxiv: math.NT/0604232.
- 3.
- M. Einsiedler and A. Katok.
Rigidity of measures - the high entropy case, and non-commuting foliations. Israel J. Math., 148 (2005), 169-238. MR 2191228 (2007d:37034)
- 4.
- M. Einsiedler and A. Katok.
Invariant measures on for split simple Lie-groups . Comm. Pure Appl. Math., 56 (8) (2003), 1184-1221. MR 1989231 (2004e:37042)
- 5.
- M. Einsiedler, A. Katok, and E. Lindenstrauss.
Invariant measures and the set of exceptions to Littlewood's conjecture. Ann. of Math. (2), 164 (2006), no. 2, 513-560. MR 2247967
- 6.
- Manfred Einsiedler and Elon Lindenstrauss.
Diagonalizable flows on locally homogeneous spaces and number theory. Proceedings of the International Congress of Mathematicians. International Congress of Mathematicians. Vol. II, 1731-1759, Eur. Math. Soc., Zürich, 2006. 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). MR 2122918 (2005k:37008)
- 9.
- Anatole Katok and Ralf Spatzier.
Invariant measures for higher-rank hyperbolic abelian actions. Ergodic Theory Dynam. Systems, 16 (1996), no. 4, 751-778. MR 1406432 (97d:58116)
- 10.
-
Corrections to: ``Invariant measures for higher-rank hyperbolic abelian actions''. Ergodic Theory Dynam. Systems, 18 (1998), no. 2, 503-507. MR 1619571 (99c:58093)
- 11.
- Y. Katznelson.
Chromatic numbers of Cayley graphs on and recurrence. Combinatorica, 21 (2001). MR 1832446 (2002h:05065)
- 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 (87i:58101a)
- 13.
- F. Ledrappier and L.-S. Young.
The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. of Math. (2), 122 (1985), no. 3, 540-574. MR 819557 (87i:58101b)
- 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. Annals of Math. (2), 163 (2006). MR 2195133 (2007b:11072)
- 16.
- Elon Lindenstrauss.
Rigidity of multiparameter actions. Israel J. of Math., 149 (2005). MR 2191215 (2006j:37007)
- 17.
- Gregory Margulis.
Problems and conjectures in rigidity theory. In Mathematics: Frontiers and perspectives, pages 161-174. Amer. Math. Soc., Providence, RI, 2000. MR 1754775 (2001d:22008)
- 18.
- G. A. Margulis and G. Tomanov.
Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. Invent. Math., 116 (1994), nos. 1-3, 347-392. MR 1253197 (95k:22013)
- 19.
- Andrew Pollington and Sanju Velani.
On a problem in simultaneous Diophantine approximation: Littlewood's conjecture. Acta. Math., 185 (2000). MR 1819996 (2002a:11076)
- 20.
- M. Ratner.
Horocycle flows, joinings and rigidity of products. Ann. of Math. (2), 118 (2) (1983), 277-313. MR 717825 (85k:58063)
- 21.
- Marina Ratner.
On Raghunathan's measure conjecture. Ann. of Math. (2), 134 (3) (1991), 545-607. MR 1135878 (93a:22009)
- 22.
- Marina Ratner.
Raghunathan's topological conjecture and distributions of unipotent flows. Duke Math. J., 63 (1) (1991), 235-280. MR 1106945 (93f:22012)
- 23.
- Carl Siegel.
Lectures on the geometry of numbers. Springer-Verlag, Berlin, 1989. MR 1020761 (91d:11070)
- 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.
- P. Walters.
An introduction to ergodic theory. Graduate Texts in Mathematics, 79. Springer-Verlag, 1982. MR 648108 (84e:28017)
- 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:
http://dx.doi.org/10.1090/S0273-0979-07-01194-9
PII:
S 0273-0979(07)01194-9
Received by editor(s):
May 11, 2007
Received by editor(s) in revised form:
May 28, 2007
Posted:
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.
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