EDIT: Note: Some of what I wrote in this post is false. The metric on the set $S$ in Bhargava's paper has to be defined by $d\left(s,t\right) = -v_p\left(s,t\right)$ (not by $d\left(s,t\right) = p^{-v_p\left(s,t\right)}$) in order for the $p$-orderings to be the greedy $\infty$-simplices. This requires slightly generalizing the definition of a metric: Instead of being a map to $\mathbb{R}_{\geq 0}$, it now has to be a map to $\mathbb{R} \cup \left\{-\infty\right\}$ which takes the value $-\infty$ only at pairs of the form $\left(a,a\right)$. This is no longer a metric in the proper sense of this word, but just a symmetric map on $U \times U$ that satisfies $d\left(x, z\right) \leq \max\left\{d\left(x,y\right), d\left(y,z\right)\right\}$. Soon, a preprint by Fedor Petrov and myself will come out which discusses these issues in greater detail.
