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We develop the classical theory of Diophantine approximation without assumingmonotonicity or convexity. A complete `multiplicative' zero-one law isestablished akin to the `simultaneous' zero-one laws of Cassels and Gallagher.As a consequence we are able to establish the analogue of the Duffin-Schaeffertheorem within the multiplicative setup. The key ingredient is the rathersimple but nevertheless versatile `cross fibering principle'. In a nutshell itenables us to `lift' zero-one laws to higher dimensions.

Multiplicative zero-one laws and metric number theory