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-1/2. 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 (ξ₁,ξ₂)² satisfies the inhomogeneous Littlewood conjecture if q→∞q\|qξ₁-θ₁\|Z\|qξ₂-θ₂\|Z=0 for all (θ₁,θ₂)², where \|·\|Z denotes the distance to the nearest integer. We prove that the Hausdorff dimension of the set of pairs (ξ₁,ξ₂)² not satisfying the inhomogeneous Littlewood conjecture is $1$, which is equal to the Hausdorff dimension of the conjectural set of exceptions.