On divergent on average trajectories for higher rank actions (2403.16559v2)

Abstract: 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.