On The Hausdorff Dimension of Weighted Badly Approximable Vectors
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917402676035584 |
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| author | Lou, Yi |
| author_facet | Lou, Yi |
| contents | Let $\boldsymbolτ=(τ_1,\dots,τ_m)\in \mathbb{R}_{\ge 0}^m$ satisfy $\sum_{i=1}^m τ_i>1$ and $τ_1\ge \cdots \ge τ_m$ Let $Ψ_{\boldsymbolτ}=(ψ_1,\dots,ψ_m)$ be given by $$ ψ_i(q)=q^{-τ_i}, \qquad i=1,\dots,m,$$ and denote by $\mathcal{A}_m(Ψ_{\boldsymbolτ})$ the set of $Ψ_{\boldsymbolτ}$-approximable vectors in $[0,1]^m$. The associated set of weighted $Ψ_{\boldsymbolτ}$-badly approximable vectors is defined by $$\mathcal{B}_m(Ψ_{\boldsymbolτ}) = \mathcal{A}_m(Ψ_{\boldsymbolτ}) \setminus \bigcap_{0<c<1}\mathcal{A}_m(cΨ_{\boldsymbolτ}).$$ The main result of this paper is that, for every ball $B\subseteq [0,1]^m$, \[ \dim_{\mathcal{H}}\bigl(B\cap \mathcal{B}_m(Ψ_{\boldsymbolτ})\bigr) = \dim_{\mathcal{H}}\mathcal{A}_m(Ψ_{\boldsymbolτ}). \] 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_06137 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On The Hausdorff Dimension of Weighted Badly Approximable Vectors Lou, Yi Number Theory 11J13 Let $\boldsymbolτ=(τ_1,\dots,τ_m)\in \mathbb{R}_{\ge 0}^m$ satisfy $\sum_{i=1}^m τ_i>1$ and $τ_1\ge \cdots \ge τ_m$ Let $Ψ_{\boldsymbolτ}=(ψ_1,\dots,ψ_m)$ be given by $$ ψ_i(q)=q^{-τ_i}, \qquad i=1,\dots,m,$$ and denote by $\mathcal{A}_m(Ψ_{\boldsymbolτ})$ the set of $Ψ_{\boldsymbolτ}$-approximable vectors in $[0,1]^m$. The associated set of weighted $Ψ_{\boldsymbolτ}$-badly approximable vectors is defined by $$\mathcal{B}_m(Ψ_{\boldsymbolτ}) = \mathcal{A}_m(Ψ_{\boldsymbolτ}) \setminus \bigcap_{0<c<1}\mathcal{A}_m(cΨ_{\boldsymbolτ}).$$ The main result of this paper is that, for every ball $B\subseteq [0,1]^m$, \[ \dim_{\mathcal{H}}\bigl(B\cap \mathcal{B}_m(Ψ_{\boldsymbolτ})\bigr) = \dim_{\mathcal{H}}\mathcal{A}_m(Ψ_{\boldsymbolτ}). \] 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. |
| title | On The Hausdorff Dimension of Weighted Badly Approximable Vectors |
| topic | Number Theory 11J13 |
| url | https://arxiv.org/abs/2603.06137 |