Schmidt's theorem, Hausdorff measures and Slicing

Victor Beresnevich; Sanju Velani

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Schmidt's theorem, Hausdorff measures and Slicing

Victor Beresnevich·Sanju Velani

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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 [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.

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Victor Beresnevich, Sanju Velani. 2005-07-18. Schmidt's theorem, Hausdorff measures and Slicing. https://arxiv.org/abs/math/0507369

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