Littlewood and Duffin--Schaeffer-type problems in diophantine approximation.

Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully-inhomogeneous version of Gallagher's theorem, a diophantine fibre refinement, and a sharp and unexpected threshold for Liouville fibres. Along the way, we prove an inhomogeneous version of the Duffin--Schaeffer conjecture for a class of non-monotonic approximation functions.