There is no analogue to Jarn\'ik's relation for twisted Diophantine approximmation

TL;DR

This paper investigates the possibility of extending Jarník's relation to higher dimensions and the multiplicative case, concluding that no such generalization exists in a twisted approximation setting.

Contribution

The paper introduces a twisted Diophantine approximation setting and demonstrates that Jarník's relation has no analogue in this context.

Findings

No generalization of Jarník's relation in higher dimensions

The twisted case differs fundamentally from classical and multiplicative cases

Provides insight into the limitations of extending Diophantine relations

Abstract

Jarn\'ik's relation is in dimension $2$ the formula $\hat{\lambda} + \frac{1}{\hat{\omega}} = 1$ linking both uniform exponents. It is an open question to generalize this equation to higher dimension, or to the multiplicative case. In this paper we consider a twisted case, between the classical and the multiplicative one, and we show that no analogue to Jarn\'ik's relation holds.