On transference principle and Nesterenko's linear independence criterion

TL;DR

This paper explores transference inequalities in Diophantine approximation, offers a simple geometric proof, derives new corollaries, and simplifies Nesterenko's linear independence criterion.

Contribution

It provides a straightforward geometric proof of Schmidt and Summerer's transference inequalities and derives new corollaries, including connections to other known inequalities and a simplified proof of Nesterenko's criterion.

Findings

Proved Schmidt and Summerer's transference inequalities using geometric observations.

Derived new corollaries linking various Diophantine inequalities.

Simplified the proof of Nesterenko's linear independence criterion.

Abstract

We consider the problem of simultaneous approximation of real numbers $\theta_1, \ldots,\theta_n$ with rationals and the dual problem of approximating zero with the values of the linear form $x_0+\theta_1x_1+\ldots+\theta_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation, which proves their result. We also derive several corollaries previously unknown. Particularly, we show that, together with the transference inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalities by Bugeaud and Laurent and "one half" of the inequalities by Marnat and Moshchevitin. Besides that, we show that our main construction provides a rather simple proof of Nesterenko's linear independence criterion.