On a question of Schmidt and Summerer concerning $3$-systems

TL;DR

This paper constructs specific 3-systems and n-systems in Diophantine approximation with a property $ar{}_n=1$, generalizing previous suggestions and providing explicit examples and visualizations.

Contribution

It introduces a method to construct n-systems with $ar{}_n=1$ for any $n \\geq 3$, extending prior work and offering explicit examples and visualizations.

Findings

Constructed a proper 3-system with $ar{}_3=1$

Generalized the construction to n-systems for any n ≥ 3

Provided explicit examples and visualizations of the systems

Abstract

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.