On transference principle and Nesterenko's linear independence criterion

2021/03/22 by Oleg N. German, Nikolay Moshchevitin, German, Oleg N. +1
Mathematics · #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Equations Stability Results #Mathematical Analysis and Transform Methods #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2103.12113

openalex publication_date 2021/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem of simultaneous approximation of real numbers θ₁, …,θ_n with rationals and the dual problem of approximating zero with the values of the linear form x₀+θ₁x₁+…+θ_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.

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