Counter-examples in parametric geometry of numbers
Volume 196 / 2020
Acta Arithmetica 196 (2020), 303-323 MSC: Primary 11J13; Secondary 11J82. DOI: 10.4064/aa191217-9-4 Published online: 3 July 2020
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 semialgebraic and closed under coordinatewise minimum for $n\le 3$. In this paper, we give examples showing that for $n\ge 4$ each of the latter properties may fail.