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A Frostman-Type Lemma for Sets with Large Intersections, and an Application to Diophantine Approximation

Part of: Diophantine approximation, transcendental number theory Classical measure theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  27 October 2014

Tomas Persson  and
Henry W. J. Reeve
Tomas Persson
Affiliation:
Centre for Mathematical Sciences, Lund University, Box 118, SE-22100 Lund, Sweden, (tomasp@maths.lth.se)
Henry W. J. Reeve
Affiliation:
Department of Mathematics, University of Bristol, University Walk, Clifton, Bristol BS8 1TW, UK, (henrywjreeve@gmail.com)
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Abstract

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We consider classes of subsets of [0, 1], originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least s. We provide a Frostman-type lemma to determine if a limsup set is in such a class. Suppose that E = lim sup En ⊂ [0, 1], and that μn are probability measures with support in En. If there exists a constant C such that

for all n, then, under suitable conditions on the limit measure of the sequence (μn), we prove that the set E is in the class .

As an application we prove that, for α > 1 and almost all λ ∈ (½, 1), the set

where and ak ∈ {0, 1}}, belongs to the class . This improves one of our previously published results.

Keywords

diophantine approximationHausdorff dimensionclasses with large intersections

MSC classification

Primary: 11J83: Metric theory
Secondary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension 28A78: Hausdorff and packing measures 28A80: Fractals

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 2 , June 2015 , pp. 521 - 542
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

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