Invariant measures and the set of exceptions to Littlewood's conjecture
  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. 
Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss