Hausdorff dimension for exactly approximable matrices

We establish the Hausdorff dimension for exactly approximable matrices for all dimensions, including the inhomogeneous case. More precisely, let $\psi :\mathbb{N}\to (0,\mathrm{\infty })$ be a non-increasing approximation function and let ${\tau }_{\psi }$ denote its lower order. For $m,n\ge 1$ and $\mathit{\theta }\in {\mathbb{R}}^m$, denote ${\text{Exact}}_{m,n}(\psi ,\mathit{\theta })$ the set of $m\times n$ matrices that are $\psi$-approximable with shift $\mathit{\theta }$, but are not $c\psi$-approximable for any $0<c<1$. It is shown that $$\text{hdim}{\text{Exact}}_{m,n}(\psi ,\mathit{\theta })=m(n-1)+\frac{m+n}{1+{\tau }_{\psi }},\text{ for }{\tau }_{\psi }>\frac{n}{m}\mathrm{.}$$ We adopt a probabilistic viewpoint, i.e. by an averaged counting argument to quantify the distribution of rational vectors and subspaces $$\{\frac{\mathbf{p}+\mathit{\theta }}{q}:(\mathbf{p},q)\in {\mathbb{Z}}^{m+1}\},\left\{\right\{A\in {\text{Mat}}_{m,n}(\mathbb{R}):A\mathbf{q}-\mathbf{p}-\mathit{\theta }=0\}:(\mathbf{p},\mathbf{q})\in {\mathbb{Z}}^{m+n}\}\mathrm{.}$$