A Mass Transference Principle and the Duffin–Schaeffer conjecture for Hausdorff measures

2006/11/01 by Victor Beresnevich, Sanju Velani · 7 citations
Mathematics · #Mathematical Dynamics and Fractals #Advanced Topology and Set Theory #Functional Equations Stability Results #Mathematics #Conjecture #Hausdorff space #Hausdorff measure #Pure mathematics #Hausdorff dimension

paper · pdf · doi:10.4007/annals.2006.164.971

openalex publication_date 2006/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

Abstract

A Hausdorff measure version of the Duffin-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 lim sup subsets of R k to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of lim sup 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.

Citations

• Fractal Geometry: Mathematical Foundations and Applications.
• Khintchine’s problem in metric Diophantine approximation

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