On The Hausdorff Dimension of Weighted Badly Approximable Vectors

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.