On transfer inequalities in Diophantine approximation, II

TL;DR

This paper refines the Khintchine Transference Principle in Diophantine approximation by incorporating additional uniform exponents, leading to sharper inequalities relating rational approximation and linear independence.

Contribution

It introduces improved bounds for transfer inequalities in Diophantine approximation by considering two additional uniform exponents, enhancing the classical Khintchine inequalities.

Findings

Refined Khintchine Transference Principle incorporating two uniform exponents

Sharpened inequalities relating rational approximation and linear independence

Optimality of classical inequalities maintained, with improvements in bounds

Abstract

We refine Khintchine Transference Principle which relates the measure of simultaneous rational approximation of an $n$ real numbers with the measure of linear independence of these $n$ numbers. Khintchine's inequalities are known to be optimal. However, they may be sharpened by taking into account two further uniform exponents.