A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures

Victor Beresnevich; Sanju Velani

doi:10.4007/annals.2006.164.971

A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures

Victor Beresnevich, Sanju Velani

Mathematics

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

Abstract

A Hausdorff measure version of the Dufin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup subsets of Rk to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of limsup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

Original language	English
Pages (from-to)	971-992
Number of pages	21
Journal	Annals of Mathematics
Volume	164
Issue number	3
DOIs	https://doi.org/10.4007/annals.2006.164.971
Publication status	Published - Nov 2006

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10.4007/annals.2006.164.971

http://annals.princeton.edu/annals/2006/164-3/p06.xhtml

Cite this

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title = "A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures",

abstract = "A Hausdorff measure version of the Dufin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup subsets of Rk to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of limsup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.",

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AU - Beresnevich, Victor

AU - Velani, Sanju

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N2 - A Hausdorff measure version of the Dufin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup subsets of Rk to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of limsup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

AB - A Hausdorff measure version of the Dufin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup subsets of Rk to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of limsup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

U2 - 10.4007/annals.2006.164.971

DO - 10.4007/annals.2006.164.971

M3 - Article

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VL - 164

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EP - 992

JO - Annals of Mathematics

JF - Annals of Mathematics

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