Counter-examples in Parametric Geometry of Numbers

TL;DR

This paper explores the properties of spectra of Diophantine approximation exponents in parametric geometry of numbers, showing that certain properties hold for low dimensions but can fail in higher dimensions.

Contribution

It provides explicit counterexamples demonstrating that the spectra are not always semi-algebraic or closed under coordinate-wise minimum for dimensions four and above.

Findings

Spectra are semi-algebraic for n ≤ 3

Spectra are closed under coordinate-wise minimum for n ≤ 3

Counterexamples show failure of these properties for n ≥ 4

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.