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# Article

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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:
[1] Laurent, M.: Exponents of Diophantine approximation in dimension two. Canadian Journal of Mathematics, 61, 1, 2009, 165-189, Cambridge University Press, DOI 10.4153/CJM-2009-008-2 | MR 2488454
[2] Roy, D.: On Schmidt and Summerer parametric geometry of numbers. Annals of Mathematics, 2015, 739-786, JSTOR, DOI 10.4007/annals.2015.182.2.9 | MR 3418530
[3] Roy, D.: On the topology of Diophantine approximation spectra. Compositio Mathematica, 153, 7, 2017, 1512-1546, London Mathematical Society, DOI 10.1112/S0010437X17007126 | MR 3705265
[4] Schleischitz, J.: Diophantine approximation and special Liouville numbers. Communications in Mathematics, 21, 1, 2013, 39-76, MR 3067121
[5] Schleischitz, J.: On approximation constants for Liouville numbers. Glasnik matematički, 50, 2, 2015, 349-361, Hrvatsko matematičko društvo i PMF-Matematički odjel, Sveučilišta u Zagrebu,
[6] Schmidt, W.M., Summerer, L.: Parametric geometry of numbers and applications. Acta Arithmetica, 140, 2009, 67-91, Instytut Matematyczny Polskiej Akademii Nauk, Zbl 1236.11060
[7] Schmidt, W.M., Summerer, L.: Diophantine approximation and parametric geometry of numbers. Monatshefte für Mathematik, 169, 1, 2013, 51-104, Springer, DOI 10.1007/s00605-012-0391-z | MR 3016519
[8] Schmidt, W.M., Summerer, L.: Simultaneous approximation to three numbers. Moscow Journal of Combinatorics and Number Theory, 3, 1, 2013, 84-107, MR 3284111
[9] Schmidt, W.M., Summerer, L.: Simultaneous approximation to two reals: bounds for the second successive minimum. Mathematika, 63, 3, 2017, 1136-1151, Wiley Online Library, DOI 10.1112/S0025579317000274 | MR 3731318

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