2005/07/18 by Victor Beresnevich, Beresnevich, Victor, Sanju Velani +1 · 1 citation
Mathematics · #11J83 #28A78 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11J83 #msc:28A78
paper · pdf · doi:10.48550/arxiv.math/0507369
20 pages
arxiv created 2005/07/18 · arxiv updated 2009/12/01
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.