  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.
