  Thanks to recent advances in parametric geometry of numbers, we know that the
spectrum of any set of $m$ exponents of Diophantine approximation to points in
$\mathbb{R}^n$ (in a general abstract setting) is a compact connected subset of
$\mathbb{R}^m$. Moreover, this set is semi-algebraic and closed under
coordinate-wise minimum for $n\le 3$. In this paper, we give examples showing
that for $n\ge 4$ each of the latter properties may fail.
