Multiplicative zero-one laws and metric number theory

We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete multiplicative' zero-one law is established akin to the simultaneous’ zero-one laws of Cassels and Gallagher. As a consequence we are able to establish the analogue of the Duffin-Schaeffer theorem within the multiplicative setup. The key ingredient is the rather simple but nevertheless versatile cross fibering principle'. In a nutshell it enables us to lift’ zero-one laws to higher dimensions.