A geometric proof of Jarnik's identity in the setting of weighted simultaneous approximation

TL;DR

This paper offers a geometric proof of a weighted version of Jarnik's identity in simultaneous approximation, using parametric geometry of numbers to replace previous classical methods.

Contribution

It introduces a new geometric approach based on successive minima functions to generalize Jarnik's identity to the weighted setting.

Findings

Provides a geometric proof of weighted Jarnik's identity

Extends the identity to the weighted simultaneous approximation setting

Demonstrates the effectiveness of parametric geometry of numbers in this context

Abstract

Jarnik's identity plays a major role in classical simultaneous approximation to two real numbers. O. German [2] has shown a generalization to the weighted setting in which the identity has to be replaced by two inequalities. His methods belong to classical geometry of numbers. The aim of this paper is to provide an alternative approach based on a careful examination of certain successive minima functions that stem from parametric geometry of numbers, a method that has already been successfully employed to generalize Jarnik's identity to higher dimensions in the classical setup in [3] and [7].