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

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].