For $d\ge 3$ 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 $\frac{ d-1}{ 2}$. 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 $(\xi_1,\xi_2)\in\mathbb{ R}^2$ satisfies the inhomogeneous Littlewood conjecture if $$\liminf_{ q\to\infty}q\|q\xi_1-\theta_1\|_{ \mathbb{ Z}}\|q\xi_2-\theta_2\|_{ \mathbb{ Z}}=0$$ for all $(\theta_1,\theta_2)\in\mathbb{ R}^2$, where $\|\cdot\|_\mathbb{ Z}$ denotes the distance to the nearest integer. We prove that the Hausdorff dimension of the set of pairs $(\xi_1,\xi_2)\in\mathbb{ R}^2$ not satisfying the inhomogeneous Littlewood conjecture is $1$, which is equal to the Hausdorff dimension of the conjectural set of exceptions.
Categories: math.DS math.NT