On The Hausdorff Dimension of Weighted Badly Approximable Vectors

TL;DR

This paper determines the Hausdorff dimension of weighted badly approximable vectors, extending previous unweighted results using Cantor-type constructions and mass distribution methods.

Contribution

It generalizes the Hausdorff dimension results for badly approximable vectors to the weighted setting, independent of recent weighted approximation results.

Findings

Hausdorff dimension of weighted badly approximable vectors equals that of the approximable set.

Extends Cantor-type and mass distribution techniques to weighted approximation.

Results hold uniformly over all balls in the unit cube.

Abstract

Let $\boldsymbol{\tau}=(\tau_1,\dots,\tau_m)\in \mathbb{R}_{\ge 0}^m$ satisfy $\sum_{i=1}^m \tau_i>1$ and $\tau_1\ge \cdots \ge \tau_m$ Let $\Psi_{\boldsymbol\tau}=(\psi_1,\dots,\psi_m)$ be given by $$ \psi_i(q)=q^{-\tau_i}, \qquad i=1,\dots,m,$$ and denote by $\mathcal{A}_m(\Psi_{\boldsymbol\tau})$ the set of $\Psi_{\boldsymbol\tau}$-approximable vectors in $[0,1]^m$. The associated set of weighted $\Psi_{\boldsymbol\tau}$-badly approximable vectors is defined by $$\mathcal{B}_m(\Psi_{\boldsymbol\tau}) = \mathcal{A}_m(\Psi_{\boldsymbol\tau}) \setminus \bigcap_{0<c<1}\mathcal{A}_m(c\Psi_{\boldsymbol\tau}).$$ The main result of this paper is that, for every ball $B\subseteq [0,1]^m$, \[ \dim_{\mathcal{H}}\bigl(B\cap \mathcal{B}_m(\Psi_{\boldsymbol\tau})\bigr) = \dim_{\mathcal{H}}\mathcal{A}_m(\Psi_{\boldsymbol\tau}). \] The proof extends the Cantor-type construction and mass distribution arguments of Koivusalo, Levesley, Ward, and Zhang from the unweighted to the weighted setting, and is independent of recent results on weighted exact approximation.