Hausdorff dimension for exactly approximable matrices

Abstract

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