On the transference principle and Nesterenko's linear independence criterion

Abstract

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 $x_0+\theta_1x_1+…+\theta_nx_n$ 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 previously unknown corollaries. In particular, we show that, together with German's 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. Moreover, we show that our main construction provides a rather simple proof of Nesterenko's linear independence criterion.