On Diophantine exponents and Khintchine's transference principle

TL;DR

This paper advances the understanding of Diophantine exponents by refining existing estimates for transposed systems and generalizing to arbitrary systems, also improving constants in Mahler's transference theorem.

Contribution

It improves estimates for uniform Diophantine exponents and generalizes bounds for individual exponents to arbitrary systems, enhancing Mahler's transference theorem.

Findings

Refined estimates of Jarnik and Apfelbeck for uniform exponents.

Generalized Laurent and Bugeaud's bounds to arbitrary systems.

Provided a better constant in Mahler's transference theorem.

Abstract

In this paper we improve estimates of Jarnik and Apfelbeck for uniform Diophantine exponents of transposed systems of linear forms and generalize to the case of an arbitrary system the estimates of Laurent and Bugeaud for individual exponents. The method proposed also gives a better constant in Mahler's transference theorem.