Talk:Integers with Metric Induced by P-adic Valuation

Equivalent Definitions

The definitions

Definition 1

Let $p \in \N$ be a prime.

Let $\norm {\cdot}_p: \Z \to \R_{\ge 0}$ be the restriction to $\Z$ of the $p$-adic norm on $\Q$.

The restricted $p$-adic metric on $\Z$ is the metric induced by $\norm {\cdot}_p$:

$\forall x, y \in \Z: \map d {x, y} = \norm {x - y}_p$

Definition 2

Let $p \in \N$ be a prime.

Let $d: \Z^2 \to \R_{\ge 0}$ be the mapping defined as:

$\forall x, y \in \Z: \map d {x, y} = \begin {cases} 0 & : x = y \\ \dfrac 1 r & : x - y = p^{r - 1} k: r, k \in \Z, p \nmid k \end {cases}$

Then $d$ is known as the restricted $p$-adic metric on $\Z$.

(1975: W.A. Sutherland: Introduction to Metric and Topological Spaces: $2$: Continuity generalized: metric spaces: Exercise $2.6: 23$)

are not equivalent. They are both metrics. They both define the same topology. Indeed they define the same set of open balls. The two definitions are related, but not the same. Consider the distance between $p^2$ and $0$.

According to Definition 1:

$\map d {p^2, 0} = \dfrac 1 {p^2}$

According to Definition 2:

$\map d {p^2, 0} = \dfrac 1 3$

The version of Definition 2 that is equivalent to Definition 1 is the following:

Definition 2

Let $p \in \N$ be a prime.

Let $d: \Z^2 \to \R_{\ge 0}$ be the mapping defined as:

$\forall x, y \in \Z: \map d {x, y} = \begin {cases} 0 & : x = y \\ \dfrac 1 {p^r} & : x - y = p^r k: r, k \in \Z, p \nmid k \end {cases}$

Then $d$ is known as the restricted $p$-adic metric on $\Z$.

In edition 2 of WA Sutherland: Introduction to Metric & Topological Spaces, Chapter 5 Metric Spaces, Example 5.11, Sutherland does not refer to Definition 2 as the $p$-adic metric. --Leigh.Samphier (talk) 09:40, 7 April 2021 (UTC)

... hmm ... what would you call it then? Looks like I have work to do. Afraid I haven't seen edition 2 of Sutherland, all I have is my 1975 Edition 1 from when I was first studying. --prime mover (talk) 10:50, 7 April 2021 (UTC)

I don't know what the WA Sutherland metric is called. The metric is the inverse of the $p$-adic valuation plus one, that is:

if $\nu_p: \Z \to \N \cup \set {+\infty}$ is the $p$-adic valuation then:

$\forall x, y \in \Z: \map d {x, y} = \dfrac 1 {\map {\nu_p}{x-y} + 1}$

Its probably the case that every valuation will induce a metric this way. The proof in WA Sutherland is not specific to the $p$-adic valuation. A cursory search didn't find anything that discussed this metric from a valuation. the induced metric from a valuation was more along the lines of the $p$-adic metric of definition 1. So I cant reach any conclusion as to what to call the metric from Sutherland. Its some sort of metric induced by the $p$-adic valuation. It doesn't appear to be a common metric. --Leigh.Samphier (talk) 11:56, 7 April 2021 (UTC)

Sutherland does not call it anything. He just describes it, and sets the exercise to prove it's a metric. Without looking at it quite closely enough, I thought it was the same thing, so I appended it as a second definition, meaning to come back to it later when I wasn't tired of the subject. --prime mover (talk) 14:03, 7 April 2021 (UTC)

So Definition 2 should be a theorem that proves that the defined function is a metric. The theorem should mention that this metric is not to be confused with the $p$-adic metric which is another metric derived from the $p$-adic valuation.

Is Definition 1 needed? I think it isn't. --Leigh.Samphier (talk) 10:16, 8 April 2021 (UTC)

"Is Definition 1 needed?" Good question. I appreciate that Definition 2 defines an un-named metric which was invented by Sutherland for the purposes of providing an exercise to be worked by the student. As such the exercise can be rewritten by defining the metric internal to the Result page that performs that proof, and the page renamed to something like Metric Space/Examples/...something arbitrary. No problem.

Now I look more closely, I see that the proof that Definition 1 is a metric comes from an exercise in Mendelson. I don't know where I got the idea for calling it the "restricted P-adic metric" but I think I just assumed that Mendelson (as I had also believed about Sutherland) was setting exercises with a purpose, perhaps as a gentle introduction to P-adic numbers, etc., and that it was all sufficiently inter-related that it may in due course have made some sense. Now I am enlightened, and it appears that these metrics are more or less arbitrary, and concocted merely as exercises to keep the student busy.

When I am once more motivated to work on metric spaces once more (I find them tedious in the extreme), I will probably revisit it. At the moment I am desperately trying to stay focused on this analytic geometry thread that I have set myself, as I want to get some background solidly in place before I am able to set myself to work on projective geometry (another big gap in our coverage). Then I will be up to speed on yet another of those areas of mathematics which was covered in my degree course, which is rapidly dwindling from my brain like sand from my ears. Getting old is suboptimal, but then again, it sure beats the alternative. --prime mover (talk) 10:53, 8 April 2021 (UTC)