On a mixed Khintchine problem in Diophantine approximation
arXiv:1201.4694
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 Jarník-Besicovich Theorem. Previous knowledge surrounding mixed problems was almost entirely restricted to the multiplicative setup of de Mathan & Teulié [21], where the concept originated.
16 pages. Updated 28/06/12 (Revised after very helpful comments from referee, resulting in a weakening of Theorem 3.1) Updated 14/02/13 (Final revision after excellent comments from second referee, upon acceptance to Moscow Journal of Combinatorics and Number Theory)