A transference principle for simultaneous rational approximation

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.