Characterization of Open Ball in P-adic Numbers

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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 open ball of center $a$ of radius $\epsilon$.


Let $n \in \Z$.

Let $x, y \in \Q_p$.


The following statements are equivalent:

$(1): \quad x \in \map {B_{p^{-n} } } y$
$(2): \quad \norm{x - y}_p < p^{-n}$
$(3): \quad \map {B_{p^{-n} } } x = \map {B_{p^{-n} } } y$
$(4): \quad x - y \in p^{n + 1} \Z_p$
$(5): \quad x + p^{n + 1} \Z_p = y + p^{n + 1} \Z_p$


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 open ball in the $p$-adic numbers.

$\Box$


Condition $(1)$ iff Condition $(3)$

By definition, $\map {B_{p^{-n}}} y$ is an open ball in a non-Archimedean norm $\norm {\,\cdot\,}_p$.

From Centers of Open 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 Open 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 $(4)$

\(\ds \norm {x - y}_p\) \(<\) \(\ds p^{-n}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \norm {x - y}_p\) \(\le\) \(\ds p^{-\paren {n + 1} }\) $p$-adic Norm of $p$-adic Number is Power of $p$
\(\ds \leadstoandfrom \ \ \) \(\ds \norm {x - y}_p\) \(\le\) \(\ds \norm {p^{n + 1} }_p\) Definition of $p$-adic Norm on Integers
\(\ds \leadstoandfrom \ \ \) \(\ds \dfrac {\norm {x - y}_p} {\norm {p^{-\paren {n + 1} } }_p}\) \(\le\) \(\ds 1\) dividing both sides by $\norm {p^{-\paren {n + 1} } }$
\(\ds \leadstoandfrom \ \ \) \(\ds \norm {p^{-\paren {n + 1} } \paren {x - y} }_p\) \(\le\) \(\ds 1\) Norm of Quotient in Division Ring
\(\ds \leadstoandfrom \ \ \) \(\ds p^{-\paren {n + 1} } \paren {x - y}\) \(\in\) \(\ds \Z_p\) Definition of $p$-adic Integers
\(\ds \leadstoandfrom \ \ \) \(\ds x - y\) \(\in\) \(\ds p^{n + 1} \Z_p\)

$\Box$


Condition $(3)$ iff Condition $(5)$

From Open Balls of P-adic Number,

$\map {B_{p^{-n} } } x = x + p^{n + 1} \Z_p$

and

$\map {B_{p^{-n} } } y = y + p^{n + 1} \Z_p$

Hence:

$\map {B_{p^{-n} } } x = \map {B_{p^{-n} } } y$ if and only if $x + p^{n + 1} \Z_p = y + p^{n + 1} \Z_p$

$\blacksquare$


Sources

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