On a mixed Khintchine problem in Diophantine approximation

TL;DR

This paper develops a new mixed version of Khintchine's theorem in Diophantine approximation, advancing the metric theory and establishing a mixed analogue of the Jarník-Besicovich Theorem using ubiquity techniques.

Contribution

It introduces a natural mixed analogue of classical Diophantine approximation theorems, extending the theory beyond the multiplicative setup and making significant progress in the metric theory.

Findings

Established a mixed Khintchine theorem in Diophantine approximation.

Developed a mixed analogue of the Jarník-Besicovich Theorem.

Applied ubiquity techniques to advance the metric theory.

Abstract

We establish a `mixed' version of a fundamental theorem of Khintchine within the field of simultaneous Diophantine approximation. Via the notion of ubiquity we are able to make significant progress towards the completion of the metric theory associated with mixed problems in this setting. This includes finding a natural mixed analogue of the classical Jarn\'ik-Besicovich Theorem. Previous knowledge surrounding mixed problems was almost entirely restricted to the multiplicative setup of de Mathan & Teuli\'e [21], where the concept originated.