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

  • Tomas Persson (a1) and Henry W. J. Reeve (a2)
  • Please note a correction has been issued for this article.

Abstract

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.

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References

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1.Bugeaud, Y., Intersective sets and Diophantine approximation, Michigan Math. J. 52 (2004), 667682.
2.Falconer, K., Classes of sets with large intersections, Mathematika 32(2) (1985), 191205.
3.Falconer, K., Sets with large intersection properties, J. Lond. Math. Soc. 49(2) (1994), 267280.
4.Palis, J. and Yoccoz, J.-C., Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension, Acta Math. 172 (1994), 91136.
5.Persson, T. and Reeve, H., On the Diophantine properties of λ-expansions, Mathematika 59(1) (2013), 6586.
6.Solomyak, B., On the random series Σ±λn (an Erdös problem), Annals Math. 142(3) (1995), 611625.
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A Frostman-Type Lemma for Sets with Large Intersections, and an Application to Diophantine Approximation

  • Tomas Persson (a1) and Henry W. J. Reeve (a2)
  • Please note a correction has been issued for this article.

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A correction has been issued for this article: