A transference principle for simultaneous rational approximation

TL;DR

This paper develops a general transference principle for irrationality measures of points with linearly independent coordinates, recovering key inequalities and providing insights into rational approximation sequences, with applications.

Contribution

It introduces a new transference principle for irrationality measures and extends existing inequalities to a broader class of points in rational approximation theory.

Findings

Established a general transference principle for irrationality measures.

Recovered an important inequality relating exponents of rational approximation.

Provided detailed information on sequences of best rational approximations near boundary cases.

Abstract

We establish a general transference principle for the irrationality measure of points with $\mathbb{Q}$-linearly independent coordinates in $\mathbb{R}^{n+1}$, for any given integer $n\geq 1$. On this basis, we recover an important inequality of Marnat and Moshchevitin which describes the spectrum of the pairs of ordinary and uniform exponents of rational approximation to those points. For points whose pair of exponents are close to the boundary in the sense that they almost realize the equality, we provide additional information about the corresponding sequence of best rational approximations. We conclude with an application.