Counter-examples in Parametric Geometry of Numbers

Abstract

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.

18 pages, 5 figures