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arXiv · 1912.06483

Counter-examples in Parametric Geometry of Numbers

Martin Rivard-Cooke·Damien Roy
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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.

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Martin Rivard-Cooke, Damien Roy. 2019-12-13. Counter-examples in Parametric Geometry of Numbers. https://arxiv.org/abs/1912.06483

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