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

Venkatesh, Akshay

doi:10.1090/S0273-0979-07-01194-9

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

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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.2307/2007030
Marina Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), no. 3, 545–607. MR 1135878, DOI https://doi.org/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 https://doi.org/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

SV Lior Silberman and Akshay Venkatesh. On quantum unique ergodicity for locally symmetric spaces. math.RT/0407413, to appear, GAFA, 17 (3) (2007), 960–998.

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.