Dimension drop for diagonalizable flows on homogeneous spaces

TL;DR

This paper proves that for a broad class of diagonalizable flows on homogeneous spaces, the set of points whose orbits miss a given open set has Hausdorff dimension strictly less than the space, confirming a long-standing conjecture.

Contribution

It establishes the conjecture for all Ad-diagonalizable flows on irreducible quotients of semisimple Lie groups, extending previous results to a more general setting.

Findings

Hausdorff dimension of orbit-missing set is smaller than space dimension

Uses exponential mixing and integral inequalities for height functions

Application to Dirichlet-Improvable systems of linear forms

Abstract

Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in $G$, let $O$ be an open subset of $X$, and let $F = \{g_t: t\ge 0\}$ be a one-parameter subsemigroup of $G$. Consider the set of points in $X$ whose $F$-orbit misses $O$; 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 is proved when $X$ is compact or when $G$ is a simple Lie group of real rank $1$, or, most recently, for certain special flows on the space of lattices. In this paper we prove this conjecture for arbitrary $\operatorname{Ad}$-diagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on $G/\Gamma$. We also derive an application to jointly Dirichlet-Improvable systems of linear forms.