On the dimension drop conjecture for diagonal flows on the space of lattices

Abstract

Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in $G$, let $U$ be an open subset of $X$, and let $\{g_t\}$ be a one-parameter subgroup of $G$. Consider the set of points in $X$ whose $g_t$-orbit misses $U$; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of $X$. This conjecture has been proved when $X$ is compact or when $G$ is a simple Lie group of real rank $1$. In this paper we prove this conjecture for the case $G=\textrm{SL}_{m+n}(\mathbb{R})$, $\Gamma=\textrm{SL}_{m+n}(\mathbb{Z})$ and $g_t=\textrm{diag} (e^{nt}, \dots, e^{nt},e^{-mt}, \dots, e^{-mt})$, in fact providing an effective estimate for the codimension. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on $\textrm{SL}_{m+n}(\mathbb{R})/\textrm{SL}_{m+n}(\mathbb{Z})$. We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation.