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

Abstract

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.

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

“…Theorem 4.3 is the main theorem of [5]. The result concerning Littlewood's conjecture is deduced from it.…”

Diagonalizable flows on locally homogeneous spaces and number theory

“…While at present this conjecture remains open, the following partial result is known: Theorem 2.2 (E., Katok, L. [14]). Let A be the group of diagonal matrices as above and n ≥ 3.…”

Rigidity of measures—The high entropy case and non-commuting foliations

“…It should be noted that for this method the properties of the lattice do not play any role, and indeed this is true not only for Γ = SL(k, Z) but for every discrete subgroup Γ. Subsequently this was called the high entropy case and the corresponding method was one of the main tools for Theorem 1.3 and the above mentioned partial result on Littlewood's conjecture [7]. A second key argument which appeared in [6] the first time was the product structure of the conditional measures.…”

“…Since the foliations under consideration in this case do commute, the methods of [6] are not applicable. This was the other method used for Theorem 1.3 and Littlewood's conjecture in [7] and was applied in the case where very few conditional measures are not δ-measures, this is the low entropy case. Here the earlier mentioned product structure of the conditional measures was proved in a more formal setting and was crucial to the argument.…”