Schmidt's theorem, Hausdorff measures, and slicing

Abstract

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 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 for simultaneous approximation and further demonstrates the surprising fact that the Lebesgue theory for lim sup sets underpins the general Hausdorff theory. 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” nonlinear Diophantine problem within the linear forms setup—the first of its kind.