  A Hausdorff measure version of W.M. Schmidt's inhomogeneous, linear forms
theorem in metric number theory is established. The key ingredient is a
`slicing' technique motivated by a standard result in geometric measure theory.
In short, `slicing' together with the Mass Transference Principle [3] allows us
to transfer Lebesgue measure theoretic statements for limsup sets associated
with linear forms to Hausdorff measure theoretic statements. This extends the
approach developed in [3] for simultaneous approximation. Furthermore, we
establish a new Mass Transference Principle which incorporates both forms of
approximation. As an application we obtain a complete metric theory for a
`fully' non-linear Diophantine problem within the linear forms setup.
  [3] V. Beresnevich and S. Velani : A Mass Transference Principle and the
Duffin--Schaeffer conjecture for Hausdorff measures, Pre-print (22pp):
arkiv:math.NT/0401118. To appear: Annals of Math.
