$x \in [0,1]$.
If in binary expansions ie series $\displaystyle x = \sum_{i=1}^{\infty} \frac{x_i}{2^i}$ where each $x_i \in \{0,1\}$ we identify the sequences $\underline{x}$ and $\underline{x}'$ by
$\underline{x} \sim \underline{x}' \iff \exists k$ such that $x_k=0,x_i=1 \ \forall i \geq k $ and $x_k'=1,x_i'=0 \ \forall i \geq k$.
Can we generalise this to ternary expansions? Note that these tails identified are exactly those one that give different binary expansions of the same number.
By ternary expansions I mean series of the form $\displaystyle x = \sum_{i=1}^{\infty} \frac{x_i}{3^i}$ where each $x_i \in \{0,1,2\}$
If the base is $b$ then the highest digit is $b-1$. If a base $b$ expansion is "terminating" in that it has a finite string of digits $n.d_1\cdots d_k,$ (the last of these being nonzero) followed by all $0$'s, then that is equivalent to the expansion obtained by subtracting $1$ from $d_k$ and appending a string of all high digits $d-1$.
To show this is an application of the sum of geometric series, and might be fun to do in detail.
So for your ternary case, such a terminating ternary as $0.0211(0000..)$ is the same as $0.0210(2222...)$ for example.