Projective metric number theory

TL;DR

This paper develops a probabilistic framework for Diophantine approximation in projective spaces over Q, proving analogues of classical theorems and conjectures using a projective metric, including the removal of monotonicity conditions.

Contribution

It extends Diophantine approximation theory to projective spaces, proving Khintchine's theorem and the Duffin-Schaeffer conjecture analogues without monotonicity constraints.

Findings

Proved the analogue of Khintchine's theorem in projective space.

Established the analogue of the Duffin-Schaeffer conjecture in higher dimensions.

Removed the monotonicity condition for finite places in the approximation theory.

Abstract

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.