On divergent on average trajectories for higher rank actions

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 $(\hat{I}{¾}_1,\hat{I}{¾}_2)\in {\mathbb{R}}^2$ satisfies the inhomogeneous Littlewood conjecture if $$\underset{q\to \infty }{\mathrm{liminf}}q\Vert q\hat{I}{¾}_1-\hat{I}{¸}_1{\Vert }_{\mathbb{Z}}\Vert q\hat{I}{¾}_2-\hat{I}{¸}_2{\Vert }_{\mathbb{Z}}=0$$ for all $(\hat{I}{¸}_1,\hat{I}{¸}_2)\in {\mathbb{R}}^2$, where $\Vert \cdot {\Vert }_{\mathbb{Z}}$ denotes the distance to the nearest integer. We prove that the Hausdorff dimension of the set of pairs $(\hat{I}{¾}_1,\hat{I}{¾}_2)\in {\mathbb{R}}^2$ not satisfying the inhomogeneous Littlewood conjecture is $1$, which is equal to the Hausdorff dimension of the conjectural set of exceptions.

51 pages, 1 figure