On divergent on average trajectories for higher rank actions

For d 3 we first show that the Hausdorff dimension of the set of A-divergent on average points in the (d-1) -dimensional closed horosphere in the space of d-dimensional Euclidean lattices, where A is the group of positive diagonal matrices, is at most d-12. In particular, this upper bound is sharp for d=3. We apply this to compute the Hausdorff dimension of the set of exceptions to the inhomogeneous uniform version of Littlewood conjecture. We say that a pair (₁, ₂) ² satisfies the inhomogeneous Littlewood conjecture if ₐq\|q₁-₁\|ₙ\|q₂-₂\|ₙ=0 for all (₁, ₂) ², where \|\|Z denotes the distance to the nearest integer. We prove that the Hausdorff dimension of the set of pairs (₁, ₂) ² not satisfying the inhomogeneous Littlewood conjecture is 1, which is equal to the Hausdorff dimension of the conjectural set of exceptions.