On the transference principle and Nesterenko's linear independence criterion

We consider the problem of simultaneous approximation of real numbers $\theta_1,…,\theta_n$ by rationals and the dual problem of approximating zero by the values of the linear form $x_0+\theta_1x_1+…+\theta_nx_n$ atinteger points. In this setting we analyse two transference inequalitiesobtained by Schmidt and Summerer. We present a rather simple geometricobservation which proves their result. We also derive several previously unknown corollaries. In particular, we show that, together with German'sinequalities for uniform exponents, Schmidt and Summerer's inequalities implythe inequalities by Bugeaud and Laurent and “one half” of the inequalitiesby Marnat and Moshchevitin. Moreover, we show that our main constructionprovides a rather simple proof of Nesterenko's linear independencecriterion.

Diophantine approximation, Diophantine exponents, transference inequalities, linear independence criterion, Diophantine approximation, Diophantine exponents, transference inequalities, linear independence criterion