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 $\psi:\N\to(0,\infty)$ be a non-increasing approximation function and let $\tau_\psi$ denote its lower order. For $m,n\ge1$ and $\boldsymbol\theta\in\mathbb R^m$, denote $\Exact_{m,n}(\psi,\boldsymbol\theta)$ the set of $m\times n$ matrices that are $\psi$-approximable with shift $\boldsymbol\theta$, but are not $c\psi$-approximable for any $0<c<1$. It is shown that \[ \hdim \Exact_{m,n}(\psi,\boldsymbol\theta)=m(n-1)+\frac{m+n}{1+\tau_\psi},\ \ \ {\text{for}}\ \tau_\psi>\frac{n}{m}. \] We adopt a probabilistic viewpoint, i.e. by an averaged counting argument to quantify the distribution of rational vectors and subspaces $$\left\{\frac{\mathbf p+\boldsymbol\theta}{q}: (\mathbf p, q)\in \mathbb{Z}^{m+1}\right\},\ \ \ \bigg\{\big\{A\in \Mat_{m, n}(\R): A\mathbf q-\mathbf p-\boldsymbol\theta=0\big\}: (\mathbf p, \mathbf q)\in \mathbb{Z}^{m+n}\bigg\}.$$