Transference inequalities for multiplicative Diophantine exponents

TL;DR

This paper establishes new inequalities for multiplicative Diophantine exponents, linking them to classical exponents and showing the equivalence of badly approximable matrices and their transposes.

Contribution

It introduces inequalities for multiplicative Diophantine exponents and proves the equivalence of badly approximable matrices and their transposes in this context.

Findings

Matrices are badly approximable if and only if their transposes are badly approximable.

Established inequalities connecting multiplicative and classical Diophantine exponents.

Abstract

In this paper we prove inequalities for multiplicative analogues of Diophantine exponents, similar to the ones known in the classical case. Particularly, we show that a matrix is badly approximable if and only if its transpose is badly approximable and establish some inequalities connecting multiplicative exponents with ordinary ones.