Dimension bound for doubly badly approximable affine forms

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 $Ba{d}_A(\epsilon )$ of $\epsilon$-badly approximable target $b$ is not full, and that for real numbers $\alpha$, ${\dim }_{H}Ba{d}_{\alpha }(\epsilon )=1$ if and only if $\alpha$ is singular on average. We show that if ${\dim }_{H}Ba{d}_A(\epsilon )=m$, then $A$ is singular on average.

20 pages. Some typos are corrected and a section on preliminaries (section 2.2) are added

References in corpus (1)

• A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation