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

TL;DR

This paper proves the dimension drop conjecture for certain diagonal flows on the space of lattices, providing effective estimates and connecting to Diophantine approximation improvements.

Contribution

It extends the conjecture's proof to non-compact cases involving SL(m+n,R)/SL(m+n,Z), using exponential mixing and integral inequalities.

Findings

Confirmed the conjecture for specific diagonal flows on lattice spaces.

Provided effective codimension estimates for the set of points with non-typical orbits.

Linked the results to improvements in Dirichlet's theorem for Diophantine approximation.

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.