# lie groups – The Hausdorff dimension of \$F^+_{m,n}\$ singular points

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?