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

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

2004-12-07

arxiv.org/abs/math/0412141v3

Ann. of Math. (2) 164 (2006), no. 3, 971--992

Mathematics

Number Theory

22 pages, published versioon

Scientific paper

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.

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