About Jarn\'ik's-type relation in higher dimension

TL;DR

This paper investigates the relationships between uniform Diophantine approximation exponents in higher dimensions, demonstrating the optimality of certain inequalities and the absence of Jarník-type relations among these exponents.

Contribution

It proves the optimality of German's transference inequalities and shows that the spectrum of uniform exponents in dimension n has non-empty interior, indicating no Jarník-type relations.

Findings

German's inequalities are optimal

Spectrum of exponents has non-empty interior

No Jarník-type relation exists among exponents

Abstract

Using the Parametric Geometry of Numbers introduced recently by W.M. Schmidt and L. Summerer and results by D. Roy, we show that German's transference inequalities between the two most classical exponents of uniform Diophantine approximation are optimal. Further, we establish that the spectrum of the $n$ uniform exponents of Diophantine approximation in dimension $n$ is a subset of $\mathbb{R}^n$ with non empty interior. Thus, no Jarn\'ik-type relation holds between them.