Non-triadic-base approximation conjecture on the Cantor set

Let $K$ be the middle-third Cantor set, let $\kappa=\operatorname{\dim_{H}}K$, let $b$ be an integer that is not a power of three, and let $v_b(\xi)$ be the approximation exponent associated with approximation by rationals with denominators powers of $b$. Define $\mathcal{V}_b(v)=\{\xi\in\mathbb{R}:v_b(\xi)\geq v\}$. Non-triadic-base approximation conjecture. Let us assume that $b$ is not a power of three. Then, for every real $\xi\in K$,

$$0\leq v_b(\xi)\leq\frac{\kappa}{1-\kappa},$$

and, for every real number $v\in[0,\kappa/(1-\kappa)]$,

$$\operatorname{\dim_{H}}(\mathcal{V}_b(v)\cap K)=\frac{1}{v+1}+\kappa-1.$$

The conjecture is presented as a probabilistic and metric analogue of the paper’s main Diophantine-approximation question for bases that are not powers of three. The paper gives heuristic support and verifies the corresponding assertion for its randomized model, but leaves the deterministic statement open.