How do p-adic fields degenerate for non-prime $p$?
Let $d(x,y)$ be the inverse of the highest power of $4$ that divides $\lvert x-y\rvert$
Then let $\Bbb Z_4$ be the completion of $\Bbb Z$ under this distance.
I expect this to be degenerate to some degree, since non-prime valuations fail to yield unique representations for numbers.
How much understanding do we have of how this degenerates? For example, are there distinct elements in $\Bbb Z$ for which $d(x,y)=0$ and can we define the infinite sequences that connect them?
What numbers, if any, will have multiple representations?
I'm asking this to better understand in general - but also with half an eye on the graph of the Collatz function, so if there is an example of how sequences are arbitrarily close in $\Bbb Z_4$, sequences of the form $S_n=4^n x+\dfrac{4^n-1}3$ which converge to $-\frac13$ and then by proxy all sequences of the form $2^n\cdot S_n:n\in\Bbb Z$, which converge to the set $-\dfrac{2^n}3$ would be the most useful examples to me.
Quite generally, completing $\Bbb Z$ with respect to an $n$-adic metric gives the direct product of the $p$-adic numbers for those prime $p$ which divide $n$: $$\Bbb Z_n := \varprojlim_{k} \Bbb Z /n^k \simeq \quad... \quad\simeq \prod_{p \vert n, \;p \text{ prime }} \Bbb Z_p$$ (fill in the missing steps with the Chinese Remainder Theorem and cofinal index sets in inverse limits).
In particular, if $n$ is the power of one prime $p$, you just get back the $p$-adic integers. One way to easily see that is by noticing that metrics $d_p$ and $d_{p^i}$ induce the same topology (even uniform structure), as each open ball of one contains an open ball of the other. (And that is a reformulation of the directed sets $(p^r)_r$ and $(p^{ir})_r$ in the inverse limits being cofinal.)
Compare also 4-adic numbers and zero divisors and Are the $p^n$-adic numbers isomorphic to the $p$-adic numbers? (where the argument might be a bit too short by making an additional assumption though, see comments).