On The Hausdorff Dimension of Weighted Badly Approximable Vectors

Let $\mathit{\tau }=({\tau }_1,\dots ,{\tau }_m)\in {\mathbb{R}}_{\ge 0}^m$ satisfy ${\sum }_{i=1}^m{\tau }_i>1$ and ${\tau }_1\ge \cdots \ge {\tau }_m$ Let ${\mathrm{\Psi }}_{\mathit{\tau }}=({\psi }_1,\dots ,{\psi }_m)$ be given by $${\psi }_i(q)=q^{-{\tau }_i},i=1,\dots ,m,$$ and denote by ${\mathcal{A}}_m({\mathrm{\Psi }}_{\mathit{\tau }})$ the set of ${\mathrm{\Psi }}_{\mathit{\tau }}$-approximable vectors in $[0,1{]}^m$. The associated set of weighted ${\mathrm{\Psi }}_{\mathit{\tau }}$-badly approximable vectors is defined by $${\mathcal{B}}_m({\mathrm{\Psi }}_{\mathit{\tau }})={\mathcal{A}}_m({\mathrm{\Psi }}_{\mathit{\tau }})\setminus \bigcap _{0<c<1}{\mathcal{A}}_m(c{\mathrm{\Psi }}_{\mathit{\tau }})\mathrm{.}$$ The main result of this paper is that, for every ball $B\subseteq [0,1{]}^m$, $${\dim }_{\mathcal{H}}(B\cap {\mathcal{B}}_m({\mathrm{\Psi }}_{\mathit{\tau }}))={\dim }_{\mathcal{H}}{\mathcal{A}}_m({\mathrm{\Psi }}_{\mathit{\tau }})\mathrm{.}$$ 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.