## The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture

HTML articles powered by AMS MathViewer

- by Akshay Venkatesh PDF
- Bull. Amer. Math. Soc.
**45**(2008), 117-134 Request permission

## References

- 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**
EV J. Ellenberg and A. Venkatesh. Local-global principles for representations of quadratic forms. arxiv: math.NT/0604232.
- 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**, DOI 10.1007/BF02775436 - Manfred Einsiedler and Anatole Katok,
*Invariant measures on $G/\Gamma$ 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**, DOI 10.1002/cpa.10092 - 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**, DOI 10.4007/annals.2006.164.513 - 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**
ELMV1 Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel and Akshay Venkatesh. The distribution of periodic torus orbits on homogeneous spaces. arxiv: math.DS/0607815.
- 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**, DOI 10.1017/S014338570400046X - 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**, DOI 10.1017/S0143385700009081 - 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**, DOI 10.1017/S0143385798110969 - Y. Katznelson,
*Chromatic numbers of Cayley graphs on $\Bbb Z$ and recurrence*, Combinatorica**21**(2001), no. 2, 211–219. Paul Erdős and his mathematics (Budapest, 1999). MR**1832446**, DOI 10.1007/s004930100019 - 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**, DOI 10.2307/1971328 - 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**, DOI 10.2307/1971328
EL-CDM Elon Lindenstrauss. Arithmetic quantum unique ergodicity and adelic dynamics. Proceedings of Current Developments in Mathematics conference (2004), to appear.
- Elon Lindenstrauss,
*Invariant measures and arithmetic quantum unique ergodicity*, Ann. of Math. (2)**163**(2006), no. 1, 165–219. MR**2195133**, DOI 10.4007/annals.2006.163.165 - Elon Lindenstrauss,
*Rigidity of multiparameter actions*, Israel J. Math.**149**(2005), 199–226. Probability in mathematics. MR**2191215**, DOI 10.1007/BF02772541 - Gregory Margulis,
*Problems and conjectures in rigidity theory*, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 161–174. MR**1754775** - 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**, DOI 10.1007/BF01231565 - 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**, DOI 10.1007/BF02392812 - Marina Ratner,
*Horocycle flows, joinings and rigidity of products*, Ann. of Math. (2)**118**(1983), no. 2, 277–313. MR**717825**, DOI 10.2307/2007030 - Marina Ratner,
*On Raghunathan’s measure conjecture*, Ann. of Math. (2)**134**(1991), no. 3, 545–607. MR**1135878**, DOI 10.2307/2944357 - Marina Ratner,
*Raghunathan’s topological conjecture and distributions of unipotent flows*, Duke Math. J.**63**(1991), no. 1, 235–280. MR**1106945**, DOI 10.1215/S0012-7094-91-06311-8 - 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**, DOI 10.1007/978-3-662-08287-4
SV Lior Silberman and Akshay Venkatesh. On quantum unique ergodicity for locally symmetric spaces. math.RT/0407413, to appear, - Peter Walters,
*An introduction to ergodic theory*, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR**648108**
Witte David Witte. Ratner’s theorems on unipotent flows. Chicago Lectures in Mathematics Series, University of Chicago Press, Chicago, IL, 2005.

*GAFA*, 17 (3) (2007), 960–998.

## Additional Information

**Akshay Venkatesh**- Affiliation: Department of Mathematics, Courant Institute, New York University, New York, New York 10012
- MR Author ID: 693009
- 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
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2358379