Let $G:=SL(m+n,\mathbb R)$ and $\Gamma :=SL(m+n,\mathbb Z)$ and $X:=G/\Gamma$.

(1) Let $M$ denote the set of all $m \times n$ matrices with real entries. A matrix $A \in M$ is called $\textit{singular}$ if for all $\epsilon > 0$, there exists $Q_\epsilon$ such that for all $Q \ge Q_{\epsilon}$, there exist integer vectors $p \in \mathbb Z^m$ and $q \in \mathbb Z^n$ such that

\begin{equation} \|Aq+p\|\le \epsilon Q^{-n/m} ~\text{and}~ 0<\| q \| \le Q. \end{equation}

We denote the set of singular $m\times n$ matrices by $\textbf{Sing}_{m,n}$

By Dani's correspondence principle (1985), this is equivalent to saying that $(g_t u_A \mathbb Z^n)$ is divergent in the space of unimodular lattices where $g_t:=\begin{bmatrix} e^{t/m}I_m & 0 \\ 0 & e^{-t/n}I_n \end{bmatrix}$ and $u_A:=\begin{bmatrix} I_m & A \\ 0 & I_n \end{bmatrix}$.

(2) Let $F^+:=\{g_t:t\ge 0\}$ and let $D(F^+, X)$ be the set of points $X$ such that the trajectory $F^+ x$ is divergent ("leaving any compact set").

I wonder how to show the following equation of Hausdorff dimensions:

$$\dim(X)-\dim(D(F^+, X))=mn- \dim(\textbf{Sing}_{m,n})$$

Intuitively this is very true through Dani's correspondence ("codimension"="codimension"!). But to prove it rigorously, are there any key theorems about the Hausdorff dimensions involved?