# Dimension bound for doubly badly approximable affine forms

## TL;DR

This paper establishes that for all target matrices, the set of doubly badly approximable affine forms has Hausdorff dimension less than full, linking this property to the singularity on average of the matrix.

## Contribution

It proves that the Hausdorff dimension of doubly badly approximable affine forms is not full and connects this to the singularity on average of the matrix.

## Key findings

- Hausdorff dimension of the set of doubly badly approximable matrices is not full.
- Full Hausdorff dimension implies the matrix is not singular on average.
- The result extends known properties from real numbers to matrix cases.

## Abstract

We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $\epsilon$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the Hausdorff dimension of the set $Bad_A(\epsilon)$ of $\epsilon$-badly approximable target $b$ is not full, and that for real numbers $\alpha$, $\dim_H Bad_\alpha(\epsilon)=1$ if and only if $\alpha$ is singular on average. We show that if $\dim_H Bad_A(\epsilon)=m$, then $A$ is singular on average.