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. 1. Introduction . Number theory and dynamics. There is a long and rich tradition of applying dynamical methods to number theory.