Let $K$ be the finite extension of the p-adic number field $\mathbb{Q}_p$ and let $K^{\times}$ be the set of unit elements of $K$. For a real number $r \in \mathbb{R}$,
What does the notation $r \in |K^{\times}|$ mean?
Here the absolute value $|.|$ is the p-adic absolute value.
My guess:
It may be explanation that for $a \in K^{\times}$, we have $r=|a|$. Is it?
$ |K^{\times}|$ is short notation for $\lbrace |a| : a\in K^{\times} \rbrace$.
So $r \in |K^{\times}|$ means that there is some non-zero element in $K$ which has value $r$. Another way in which this is commonly expressed is to say: $r$ is in the value group of $K$.
For example, if $K=\mathbb Q_p$ with the standard $p$-adic value (meaning $|p|= 1/p$), then
$p^{-17} \in |K^{\times}|$ and $p^{33} \in |K^{\times}|$
but
$p^{-1/5} \notin |K^{\times}|$ and $12 \notin |K^{\times}|$ and $e \notin |K^{\times}|$
although there is a finite extension $L$ of $\mathbb Q_p$ such that
$p^{-1/5} \in |L^{\times}|$;
but there is no finite extension of $\mathbb Q_p$ whose value group would contain $12$ or $e$ (unless the value is normalised differently than $|p|= 1/p$).