Infinitely Badly Approximable Affine Forms

A pair $(A,\textbf{b})$ of a real $m\times n$ matrix $A$ and $\textbf{b}\in \mathbb{R}^{m}$ is said to be infinitely badly approximable if $$ \begin{align*} & \liminf_{\textbf{q}\in\mathbb{Z}^{n}, \|\textbf{q}\|\to\infty} \|\textbf{q}\|^{\frac{n}{m}}\|A\textbf{q}-\textbf{b}\|_{\mathbb{Z}} =\infty, \end{align*} $$where $\|\cdot \|_{\mathbb{Z}}$ denotes the distance from the nearest integer vector. In this article, we introduce a novel concept of singularity for $(A,\textbf{b})$ and characterize the infinitely badly approximable property by this singular property. As an application, we compute the Hausdorff dimension of the infinitely badly approximable set. We also discuss dynamical interpretations on the space of grids in $\mathbb{R}^{m+n}$.