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

By Victor Beresnevich, Sanju Velani. 2004

264 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0c48f4f996d0cb2992d6f0f7631fbd6f349e2916

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Metric Diophantine Approximation : aspects of recent work (2016, methodology): “In the case the above s-volume sum diverges it is reasonable to believe in the truth of the following Hausdorff measure version of the Duffin-Schaeffer Conjecture [20].”
A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation (2013, methodology): “particular, we make a substantial improvement in the MTP [5] to multifractal Gibbs measures.”
Hausdorff measure of sets of Dirichlet non-improvable numbers (2017, methodology): “One can observe, however, that in proving the latter theorem the monotonicity assumption on the function x 7→ f (x)/x is only used in the last step of the proof, see the last inequality in formula (29) in [2].”
The shrinking target problem in the dynamical system of continued fractions (2014, methodology): “For this special case, the dimension of S({zn}n 1, f) follows from the theory developed by Beresnevich, Dickinson and Velani on the measure theoretical laws of limsup sets [1] (see also [2]).”
Classical metric Diophantine approximation revisited (2008, background): “It turns out that Conjecture D, a refinement of the Duffin-Schaeffer problem, is simply its consequence [ 3 ].”
On a problem of K. Mahler: Diophantine approximation and Cantor sets (2005, background): “The theorem is essentially Theorem 3 in [ 4 ].”
On the Duffin-Schaeffer conjecture (2019, background): “[4] V. Beresnevich, V. Bernik, M. Dodson and S. Velani, Classical metric Diophantine approximation revisited.”