Metric considerations concerning the mixed Littlewood Conjecture

The main goal of this note is to develop a metrical theory of Diophantine approximation within the framework of the de Mathan-Teulie Conjecture, also known as the `Mixed Littlewood Conjecture'. Let p be a prime. A consequence of our main result is that, for almost every real number \alpha, \liminf_{n\rar\infty}n(\log n)^2|n|_p\|n\alpha\|=0.