Spectrum of the exponents of best rational approximation

TL;DR

This paper applies the parametric geometry of numbers to fully characterize the spectrum of exponents related to best rational approximation in higher dimensions, extending classical transference inequalities.

Contribution

It demonstrates that the going-up and going-down transference inequalities completely describe the spectrum of approximation exponents using the new parametric geometry of numbers framework.

Findings

Full spectrum of exponents characterized by transference inequalities

Extension of classical inequalities to higher dimensions

Application of parametric geometry of numbers to Diophantine approximation

Abstract

Using the new theory of W. M. Schmidt and L. Summerer called parametric geometry of numbers, we show that the going-up and going-down transference inequalities of W. M. Schmidt and M. Laurent describe the full spectrum of the $n$ exponents of best rational approximation to points in $\mathbb{R}^{n+1}$.