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Article

MSC: 11H06, 11J13
Keywords:
parametric geometry of numbers; simultaneous approximation
Summary:
Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper $3$-system $(P_{1},P_{2},P_{3})$ with the property $\overline {\varphi }_{3}=1$. In fact, our method generalizes to provide $n$-systems with $\overline {\varphi }_{n}=1$, for arbitrary $n\geq 3$. We visualize our constructions with graphics. We further present explicit examples of numbers $\xi _{1}, \ldots , \xi _{n-1}$ that induce the $n$-systems in question.
References:
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