Dimension drop for diagonalizable flows on homogeneous spaces

Let $X = G/Γ$, where $G$ is a Lie group and $Γ$ 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 $G/Γ$. We also derive an application to jointly Dirichlet-Improvable systems of linear forms.

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