Dimension bound for doubly badly approximable affine forms

Wooyeon Kim; Seonhee Lim

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arXiv · 1904.07476

Dimension bound for doubly badly approximable affine forms

Wooyeon Kim·Seonhee Lim

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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.

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BibTeX RIS

Wooyeon Kim, Seonhee Lim. 2019-04-16. Dimension bound for doubly badly approximable affine forms. https://arxiv.org/abs/1904.07476

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