On the topology of Diophantine approximation spectra

Fix an integer $n\geqslant 2$ . To each non-zero point $\mathbf{u}$ in $\mathbb{R}^{n}$ , one attaches several numbers called exponents of Diophantine approximation . However, as Khintchine first observed, these numbers are not independent of each other. This raises the problem of describing the set of all possible values that a given family of exponents can take by varying the point $\mathbf{u}$ . To avoid trivialities, one restricts to points $\mathbf{u}$ whose coordinates are linearly independent over $\mathbb{Q}$ . The resulting set of values is called the spectrum of these exponents . We show that, in an appropriate setting, any such spectrum is a compact connected set. In the case $n=3$ , we prove moreover that it is a semi-algebraic set closed under component-wise minimum. For $n=3$ , we also obtain a description of the spectrum of the exponents $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{2},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{3})$ recently introduced by Schmidt and Summerer.

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