The exceptional set conjecture for the 3-adic Cantor set

Let $\mathbb{Z}_3$ denote the $3$-adic integers. For $\lambda\in\mathbb{Z}_3$, write

$$(\lambda)_3=(\cdots a_2a_1a_0)_3=a_0+a_1\cdot3+a_2\cdot3^2+\cdots,$$

where each $a_i\in\{0,1,2\}$. Define the $3$-adic exceptional set

$$\mathcal{E}(\mathbb{Z}_3):=\{\lambda\in\mathbb{Z}_3: \text{for infinitely many }n\geq0\text{, the expansion }(2^n\lambda)_3\text{ omits the digit }2\}.$$

Exceptional set conjecture. The $3$-adic exceptional set has Hausdorff dimension zero:

$$\dim_H\bigl(\mathcal{E}(\mathbb{Z}_3)\bigr)=0.$$

The set is forward invariant under multiplication by $2$ and is expected to be very small in measure or dimension. The paper notes that it could even be countable or equal to $\{0\}$; it also records the prior bound that its Hausdorff dimension is at most $\frac12$.