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Schmidt's theorem, Hausdorff measures and Slicing

Beresnevich, V. and Velani, S. (2006) Schmidt's theorem, Hausdorff measures and Slicing. International Mathematics Research Notices, 4879. pp. 1-24. ISSN 1687-0247

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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.

Item Type: Article
Institution: The University of York
Academic Units: The University of York > Mathematics (York)
Depositing User: York RAE Import
Date Deposited: 14 May 2009 15:34
Last Modified: 14 May 2009 15:34
Published Version: http://dx.doi.org/10.1155/IMRN/2006/48794
Status: Published
Publisher: Oxford University Press
Identification Number: 10.1155/IMRN/2006/48794
URI: http://eprints.whiterose.ac.uk/id/eprint/6412

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