The rational-point counting conjecture for Cantor sets

Let ${\mathcal{C}}$ be Cantor's middle-thirds set, and more generally let ${\mathcal{C}}$ be the set of numbers $x=\sum_{i=1}^{\infty}a_i b^{-i}$ with $a_i$ in a proper subset ${\mathcal{F}}$ of $\{0,\ldots,b-1\}$. Let $\widetilde N(T)$ be the number of purely periodic rationals $p/q\in{\mathcal{C}}$ with $q\in[(1-c)T,T]$, where $c\in(0,1)$ is fixed. Let $d$ be the Hausdorff dimension of ${\mathcal{C}}$:

$$d=\frac{\log 2}{\log 3},$$

for the ternary Cantor set, and $d=\log|{\mathcal{F}}|/\log b$ in the general case. The rational-point counting conjecture. For each $\varepsilon>0$,

$$\widetilde N(T)=O\left(T^{d+\varepsilon}\right).$$

This conjecture predicts the growth of the number of purely periodic rational points in a missing-digit fractal and is intended as a quantitative step toward intrinsic Diophantine approximation on Cantor sets. The paper presents it as a heuristic prediction supported by numerical computations; no resolution is supplied.