Projective metric number theory

In this paper we consider the probabilistic theory of Diophantine approximation in projective space over a completion of Q. Using the projective metric studied by Bombieri, van der Poorten, and Vaaler we prove the analogue of Khintchine’s Theorem in projective space. For finite places and in higher dimension, we are able to completely remove the condition of monotonicity and establish the analogue of the Duffin-Schaeffer conjecture.