An analogue of Mahler's transference theorem for multiplicative Diophantine approximation

TL;DR

This paper introduces a new analogue of Mahler's transference theorem that directly implies the multiplicative transference theorem, overcoming previous obstacles in multiplicative Diophantine approximation.

Contribution

The authors propose an analogue of Mahler's theorem tailored for the multiplicative setting, enabling a straightforward derivation of the multiplicative transference theorem.

Findings

An analogue of Mahler's theorem for multiplicative approximation is established.

The new analogue simplifies the derivation of the multiplicative transference theorem.

The approach overcomes previous obstacles in multiplicative Diophantine approximation.

Abstract

Khintchine's and Dyson's transference theorems can be very easily deduced from Mahler's transference theorem. In the multiplicative setting an obstacle appears, which does not allow deducing the multiplicative transference theorem immediately from Mahler's theorem. Some extra considerations are required, for instance, induction by the dimension. In this paper we propose an analogue of Mahler's theorem which implies the multiplicative transference theorem immediately.