Infinitely badly approximable affine forms

TL;DR

This paper introduces a new singularity concept for affine forms and characterizes infinitely badly approximable pairs, also computing their Hausdorff dimension and exploring dynamical interpretations in the space of grids.

Contribution

It presents a novel singularity notion for affine forms and links it to infinite bad approximability, providing a dimension calculation and dynamical insights.

Findings

Characterization of infinitely badly approximable pairs via a new singularity concept.

Calculation of the Hausdorff dimension of the set of such pairs.

Discussion of dynamical interpretations on the space of grids.

Abstract

A pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $\textit{infinitely badly approximable}$ if \[ \liminf_{\mathbf{q}\in\mathbb{Z}^n, \|\mathbf{q}\|\to\infty} \|\mathbf{q}\|^{\frac{n}{m}}\|A\mathbf{q}-\mathbf{b}\|_{\mathbb{Z}} =\infty, \] where $\|\cdot\|_\mathbb{Z}$ denotes the distance from the nearest integer vector. In this article, we introduce a novel concept of singularity for $(A,\mathbf{b})$ 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 $\mathbb{R}^{m+n}$.