  The parametric geometry of numbers of Schmidt and Summerer deals with
rational approximation to points in $\mathbb{R}^n$. We extend this theory to a
number field $K$ and its completion $K_w$ at a place $w$ in order to treat
approximation over $K$ to points in $K_w^n$. As a consequence, we find that
exponents of approximation over $\mathbb{Q}$ in $\mathbb{R}^n$ have the same
spectrum as their generalizations over $K$ in $K_w^n$. When $w$ has relative
degree one over a place $\ell$ of $\mathbb{Q}$, we further relate approximation
over $K$ to a point $\boldsymbol{\xi}$ in $K_w^n$, to approximation over
$\mathbb{Q}$ to a point $\Xi$ in $\mathbb{Q}_\ell^{nd}$, obtained by extension
of scalars, where $d$ is the degree of $K$ over $\mathbb{Q}$. By combination
with a result of Bel, this allows us to construct algebraic curves in
$\mathbb{R}^{3d}$ defined over $\mathbb{Q}$, of degree $2d$, containing points
that are very singular with respect to rational approximation.

Number Theory