Schmidt’s theorem, Hausdorff measures and Slicing

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 lim sup 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. 2000 Mathematics Subject Classification: Primary 11J83, 28A78; Secondary 11J13, 11K60