On a mixed Khintchine problem in Diophantine approximation

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.