On The Hausdorff Dimension Of Two Dimensional Badly Approximable Vector

Let $Ψ=(ψ_1,\dots,ψ_m)$ be an approximation function and denote by $\mathcal{A}_m(Ψ)$ the set of $Ψ$-approximable vectors in $[0,1]^m$. The associated set of weighted $Ψ$-badly approximable vectors is defined by $$\mathcal{B}_m(Ψ)=\mathcal{A}m(Ψ)\setminus\bigcap \limits{0<c<1}\mathcal{A}m(cΨ).$$ In this paper, we study the Hausdorff dimension of $\mathcal{B}m(Ψ)$ in the weighted power-law setting $ψ_i(q)=q^{-τ_i}$. Our main result establishes a sharp local Hausdorff dimension formula for $\mathcal{B}2(Ψ{\boldsymbolτ})$ when $\boldsymbolτ=(τ_1,τ_2)$ satisfies $τ_1\geqτ_2>0$ and $τ_1+τ_2>1$. We show that for any ball $B\subseteq[0,1]^2$, $$\dim{\mathcal{H}}(B\cap \mathcal{B}2(Ψ{\boldsymbol τ})) = \dim{\mathcal{H}} \mathcal{A}2(Ψ{\boldsymbol τ}) =\min \left{\frac{3+τ_1-τ_2}{1+τ_1},\frac{3}{1+τ_2}\right}.$$ 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.

💡 Research Summary

The paper investigates the Hausdorff dimension of weighted badly approximable vectors in two dimensions. For an approximation function (\Psi=(\psi_1,\dots,\psi_m)) with (\psi_i(q)=q^{-\tau_i}) (the “power‑law” case), the authors define the set of (\Psi)-approximable vectors (\mathcal A_m(\Psi)) and the associated weighted badly approximable set
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