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 $mtimes n$ 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 $F^+:={g_t:tge 0}$ 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?