Schmidt's theorem, Hausdorff measures, and slicing

Victor Beresnevich; Sanju Velani

doi:10.1155/IMRN/2006/48794

Schmidt's theorem, Hausdorff measures, and slicing

Victor Beresnevich, Sanju Velani

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

A Hausdor. 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 Hausdor. 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 Hausdor. 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.

Original language	English
Article number	48794
Pages (from-to)	1-24
Number of pages	23
Journal	International Mathematics Research Notices
Volume	48794
DOIs	https://doi.org/10.1155/IMRN/2006/48794
Publication status	Published - 2006

Access to Document

10.1155/IMRN/2006/48794

http://dx.doi.org/10.1155/IMRN/2006/48794

Cite this

@article{b1e197b25d9a42859e3a1b5c318a6cf5,

title = "Schmidt's theorem, Hausdorff measures, and slicing",

abstract = "A Hausdor. 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 Hausdor. 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 Hausdor. 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.",

author = "Victor Beresnevich and Sanju Velani",

year = "2006",

doi = "10.1155/IMRN/2006/48794",

language = "English",

volume = "48794",

pages = "1--24",

journal = "International Mathematics Research Notices",

issn = "1687-0247",

publisher = "Oxford University Press",

}

TY - JOUR

T1 - Schmidt's theorem, Hausdorff measures, and slicing

AU - Beresnevich, Victor

AU - Velani, Sanju

PY - 2006

Y1 - 2006

N2 - A Hausdor. 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 Hausdor. 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 Hausdor. 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.

AB - A Hausdor. 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 Hausdor. 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 Hausdor. 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.

U2 - 10.1155/IMRN/2006/48794

DO - 10.1155/IMRN/2006/48794

M3 - Article

SN - 1687-0247

VL - 48794

SP - 1

EP - 24

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

M1 - 48794

ER -