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On a question of Schmidt and Summerer concerning 3-systems. (English) Zbl 1475.11131

Summary: Following a suggestion of W. M. Schmidt and L. Summerer [Mathematika 63, No. 3, 1136–1151 (2017; Zbl 1388.11037)], we construct a proper 3-system \((P_1,P_2,P_3)\) with the property \(\overline{\phi}_3 = 1\). In fact, our method generalizes to provide \(n\)-systems with \(\overline{\phi}_n = 1\), for arbitrary \(n \geq 3\). We visualize our constructions with graphics. We further present explicit examples of numbers \(\xi_1, \dots,\xi_{n-1}\) that induce the \(n\)-systems in question.

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
11H06 Lattices and convex bodies (number-theoretic aspects)

Citations:

Zbl 1388.11037
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References:

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