This question is a bit vague. I was just wondering what the graph of the height function on $\mathbf{Q}$ would look like.
Define the height $h(q)$ of a rational number $q$ as follows. Write $q=a/b$, where $a$ and $b$ are coprime integers. Then $h(q) := \max(\vert a\vert,\vert b\vert)$.
The function $h$ has the property that the set of rational numbers $q$ such that $h(q)$ is bounded by a real number $C$ is finite. So I was trying to imagine how this would look like on the interval $[0,1]$ but didn't get really far. It gets arbitrarily large around any number $x \in [0,1]$.
Any thoughts?
Of course, we could also consider a height function on $\overline{\mathbf{Q}}$ and its graph.