Describe the points in $B_{d_3}(0,\frac{2}{5})$

Question

Let $d_{3}$ be the 3-adic metric above $\mathbb{Z}$ find all the points y s.t $y\in B_{d_3}(0,\frac{2}{5})$

$$B_{d_3}(0,\frac{2}{5})=\{y\in \mathbb{Z}: d_{3}(y,0)<\frac{2}{5}\}$$

So we need to find all $\frac{1}{3^{k(y)}}<\frac{2}{5}$

Where $k(y)=max\{i:3^i|y\}$

That is true for all $k(y)\neq 0$ So $y=3\mathbb{Z}$

Is this correct?

Answer 1

Yes it is. As you noted, the condition is equivalent to $k(y) \ge 1$, which translates to $3 | y$, which is equivalent to your formulation.