Einsiedler 2006 invariant measures set exceptions littlewood s

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conjecture_1_2: Littlewood's conjecture as the paper poses it: for every pair of real numbers u, v the limit inferior of n times the distances of nu and nv to the nearest integer is zero; the paper does not prove it.

corollary_1_4: States that a measure as in Theorem 1.3 is not compactly supported, and that for prime k it is the unique SL(k,R)-invariant measure on the space of unimodular lattices.

proposition_11_1: States that a real pair (u, v) satisfies liminf of n equal to zero if and only if the lattice spanned by (1,u,v), (0,1,0), (0,0,1) has an unbounded orbit under the semigroup diag(e^(-r-s), e^r, e^s), r, s > 0.

theorem_10_1: States that for k at least 3 and an open cone in the Lie algebra of the diagonal group, the lattices whose orbits under the cone stay bounded meet every unstable manifold of each element of the cone in a set of Hausdorff dimension zero.

theorem_1_3: States that for k at least 3, an ergodic measure on the space of unimodular lattices in R^k invariant under the positive diagonal group A is algebraic when some one-parameter subgroup of A acts on it with positive entropy.

theorem_1_5: States that the set of real pairs (u, v) with liminf of n positive has Hausdorff dimension zero, and is a countable union of compact sets of box dimension zero.

theorem_1_6: States that outside an A-invariant set of matrices in SL(k,R) of Hausdorff dimension k-1, the product of the k linear forms given by the rows of the matrix has infimum zero in absolute value over nonzero integer vectors.

theorem_2_1: States that for an A-invariant ergodic probability measure on SL(k,R)/Gamma, k at least 3 and Gamma discrete, each pair of indices a, b has trivial conditional measures on U_ab and U_ba, or invariance under the SL(2,R) they generate, or ergodic components on single orbits of its centralizer.

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Manfred Einsiedler, Anatole Katok, Elon Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture. Annals of Mathematics 164 (2006), 513-560. arXiv:math/0612721, doi:10.4007/annals.2006.164.513. Pages and labels cited here are those of the published article, which arXiv:math/0612721v1 reproduces with the same pagination.

The paper classifies, for k >= 3, the A-invariant and A-ergodic measures on SL(k,R)/SL(k,Z), where A is the group of positive diagonal matrices, under the hypothesis that some one-parameter subgroup of A acts with positive entropy: every such measure is algebraic (Theorem 1.3, p. 515). The paper calls this a partial result towards Margulis's Conjecture 1.1 (p. 514), which it describes as a special case of more general conjectures of Margulis and of Katok and Spatzier. The setting is the Weyl chamber flow: for k = 2 it is the geodesic flow on the modular surface, while for k > 2 the action is hyperbolic as an R^(k-1)-action, transversally to the orbits, displays rigidity properties, and A-ergodic measures are conjectured to be rare (p. 514). Theorem 2.1 (pp. 523--524), stated for any discrete subgroup Gamma, is proved for each pair of indices a, b by splitting into a high-entropy case, handled with Einsiedler and Katok's method, and a low-entropy case, handled by adapting Lindenstrauss's method with Ratner's techniques for unipotent flows (pp. 518--519, 524--525). Section 5 rules out the remaining "exceptional returns" for SL(k,Z), and Section 6 finishes with the Margulis--Tomanov measure classification.

The headline application, stated in the abstract, is "that the set of exceptions to Littlewood’s conjecture has Hausdorff dimension zero" (p. 513; Theorem 1.5, p. 516). The paper states Littlewood's conjecture as Conjecture 1.2 (p. 515), calls it well-known and long-standing, and notes that Conjecture 1.1 implies it. For problem 495, whose statement is Littlewood's conjecture, the paper's result is this dimension-zero bound on the exceptional set, not a proof of the conjecture.

Source: https://arxiv.org/abs/math/0612721. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0612721), every other right reserved.

Bears on. #495: the paper's Conjecture 1.2 is the problem's statement; Theorem 1.5 shows that the pairs violating it form a set of Hausdorff dimension zero, and Proposition 11.1 restates the problem's condition as unboundedness of an orbit in SL(3,R)/SL(3,Z). The paper does not settle the problem.

Results. Conjecture 1.2 (p. 515, Littlewood's conjecture); Theorem 1.3 (p. 515, with Margulis's Conjecture 1.1 on p. 514); Corollary 1.4 (p. 515); Theorem 1.5 (p. 516); Theorem 1.6 (p. 516, with Theorem 10.2 on p. 556); Theorem 2.1 (pp. 523--524, with Theorems 2.2 and 2.3 on p. 525); Theorem 10.1 (p. 555); Proposition 11.1 (p. 557).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.