Invariant measures and the set of exceptions to Littlewood's conjecture

By Manfred Einsiedler, A. Katok, Elon Lindenstrauss. 2006

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We classify the measures on SL (k,R)/SL (k,Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/63f1703fc7daf9ed55faf21c58675c95516187d8

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Citations in the corpus

Rigidity of measures—The high entropy case and non-commuting foliations (2005, background): “…a left action on X = Gs/F by letting an(gF) = (a(n)g)F where gF E X. Vol. 148, 2005 RIGIDITY OF MEASURES 177 Furthermore, a gives rise to an Zk-action 0 by automorphisms of Gs and an adjoint Zk-action A on gs by letting On(g) = anga -n and A n = Ads(n) for g C Gs and n E Z k, see Section 3.2.”
Invariant measures and arithmetic quantum unique ergodicity (2006, methodology): “Katok [EKL03] we show how using the methods developed in this paper in conjunction with the methods of [EK03] one can substantially sharpen the results of the latter paper.”
Multiplicatively badly approximable numbers and generalised Cantor sets (2010, background): “To date all that is know is the remarkable result of Einsiedler, Katok & Lindenstrauss [6] that states that dimMad0 = 0.”
Mesures stationnaires et fermés invariants des espaces homogènes (2011, background): “Pour νb-presque tout x dans X, on note σb,x ∈ M1(expVb) la mesure conditionnelle en x de νb pour l’action sur X du groupe expVb (voir [11] ou [5]).”
DISTRIBUTION MODULO ONE AND RATNER’S THEOREM (2006, background): “Examples are proofs of quantitative versions of the Oppenheim conjecture [3], related questions on the spacings between the values of quadratic forms [4, 8, 9], a proof of quantum unique ergodicity for certain classes of hyperbolic surfaces [6], and an approach to the Littlewood conjecture on the…”
Measure rigidity and $p$-adic Littlewood-type problems (2005, methodology): “More recently, the first named author, Katok and Lindenstrauss [EKL06] used dynamics on SL(3, R)/SL(3, Z) to show that Littlewood’s conjecture fails at most on a set of Hausdorff dimension zero.”
Distribution of periodic torus orbits and Duke's theorem for cubic fields (2007, background): “We refer the interested reader to [16] or to the original paper [15] for a historical background and for an exposition of some of the ideas that enter into the proof.”
Diagonal actions on locally homogeneous spaces (2008, methodology): “Specifically , in our joint paper with Katok [EKL06], Conjecture 1.5 is proved under the condition that there is some a ∈ A for which µ has positive entropy (see Theorem 11.5 below for a more formal statement).”
Modified Schmidt games and Diophantine approximation with weights (2008, background): “Schmidt [23] in 1982: is it true that for r = r′, the intersection of Bad(r) and Bad(r′) is nonempty? Schmidt conjectured that the answer is affirmative in the special case n = 2, r = ( (1)3 , (2)3 ) and r′ = ( (2)3 , (1)3 ), pointing out that disproving his conjecture would amount to proving Little”
The distribution of periodic torus orbits on homogeneous spaces (2006, methodology): “Using similar methods the theorem can be generalized to periodic orbits arising e.g. from cubic number fields with one real and one complex embedding (where S1 ×R>0 ∼= H PGL3(R)).”
Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows (2011, background): “These themes have arisen in the related context of measure-rigidity for algebraic partially hyperbolic actions by Einsiedler, Katok and Lindenstrauss [21].”