Baker–Schmidt-type algebraic approximation conjecture on the Cantor set

Let $K$ be the middle-third Cantor set, let $\kappa=\operatorname{\dim_{H}}K$, and let $w_n^{\ast}(\xi)$ be the exponent governing approximation of $\xi$ by algebraic numbers of degree at most $n$. Define $\mathcal{U}_n(w^{\ast})=\{\xi\in\mathbb{R}:w_n^{\ast}(\xi)\geq w^{\ast}\}$. Baker–Schmidt-type algebraic approximation conjecture. For any integer $n\geq 1$ and any real number $w^{\ast}\geq n$, the set of points in the middle-third Cantor set which are approximable at order at least $w^{\ast}+1$ by algebraic numbers of degree at most $n$ satisfies

$$\operatorname{\dim_{H}}(\mathcal{U}_n(w^{\ast})\cap K)=\max\left\{\frac{n+1}{w^{\ast}+1}+\kappa-1,\frac{\kappa}{w^{\ast}+1}\right\}.$$

This is proposed as a natural extension of the paper’s main Cantor-set dimension conjecture. The source also records an open problem asking for this Hausdorff dimension, and mentions partial results toward the conjectural formula, but does not establish the formula in full.