On the transference principle and Nesterenko’s linear independence criterion

Izvestiya. Mathematics. 2023. Vol. 87. No. 2. P. 56–68.

German O., Moshchevitin N.G.

We consider the problem of simultaneous approximation of real numbers
θ_1 ,...,θ_n by rationals and the dual problem of approximating zero by the
values of the linear form x_0 + θ_1 x_1 + ··· + θ_n x_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 partic-
ular, 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. More-
over, we show that our main construction provides a rather simple proof of
Nesterenko’s linear independence criterion.

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