Characterization of Closed Ball in P-adic Numbers
Theorem
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {{B_\epsilon}^-} a$ denote the closed ball of center $a$ of radius $\epsilon$.
Let $x, y \in \Q_p$.
Let $n \in \Z$.
The following statements are equivalent:
- $(1)\quad x \in \map {B^{\,-}_{p^{-n}}} y$
- $(2)\quad \norm{x -y}_p \le p^{-n}$
- $(3)\quad \map {B^{\,-}_{p^{-n}}} x = \map {B^{\,-}_{p^{-n}}} y$
- $(4)\quad x - y \in p^n \Z_p$
- $(5)\quad x + p^n \Z_p = y + p^n \Z_p$
Proof
From P-adic Numbers form Non-Archimedean Valued Field:
- $\norm {\,\cdot\,}_p$ is a non-Archimedean norm.
Condition $(1)$ iff Condition $(2)$
This follows directly from the definition of a closed ball in the $p$-adic numbers.
$\Box$
Condition $(1)$ iff Condition $(3)$
By definition, $\map {B^{\,-}_{p^{-n}}} y$ is a closed ball in a non-Archimedean norm $\norm {\,\cdot\,}_p$.
From Centers of Closed Balls in Non-Archimedean Division Rings:
- $x \in \map {B^{\,-}_{p^{-n}}} y \leadsto \map {B^{\,-}_{p^{-n}}} x = \map {B^{\,-}_{p^{-n}}} y$
From Center is Element of Closed Ball in P-adic Numbers:
- $\map {B^{\,-}_{p^{-n}}} x = \map {B^{\,-}_{p^{-n}}} y \leadsto x \in \map {B^{\,-}_{p^{-n}}} x = \map {B^{\,-}_{p^{-n}}} y$
$\Box$
Condition $(2)$ iff Condition $(4)$
\(\ds \norm{x - y}_p \le p^{-n}\) | \(\leadstoandfrom\) | \(\ds \norm{x - y}_p \le \norm{p^n}_p\) | Definition of $p$-adic norm on integers | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \dfrac {\norm{x - y}_p} {\norm{p^n}_p} \le 1\) | Dividing both sides of equation by $p^{-n}$ | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \norm{p^{-n} \paren{x-y} }_p \le 1\) | Norm of Quotient in Division Ring | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds p^{-n} \paren{x-y} \in \Z_p\) | Definition of $p$-adic integers | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds x-y \in p^n\Z_p\) |
$\Box$
Condition $(3)$ iff Condition $(5)$
From Closed Balls of P-adic Number,
- $\map {B^{\,-}_{p^{-n}}} x = x + p^n \Z_p$
and
- $\map {B^{\,-}_{p^{-n}}} y = y + p^n \Z_p$
Hence:
- $\map {B^{\,-}_{p^{-n}}} x = \map {B^{\,-}_{p^{-n}}} y$ if and only if $x + p^n \Z_p = y + p^n \Z_p$
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$: Lemma $3.3.5$