Đề tài "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. 1. Introduction 1.1. Number theory and dynamics. There is a long and rich tradition of applying dynamical methods to number theory.