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

Summary, in English

We consider classes G(s)([0, 1]) 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 supE(n) subset of [0, 1], and that mu(n) are probability measures with support in E-n. If there exists a constant C such that integral integral vertical bar x - y vertical bar(-s) d mu(n)(x) d mu(n)(y) < C for all n, then, under suitable conditions on the limit measure of the sequence (mu(n)), we prove that the set E is in the class G(s)([0, 1]). As an application we prove that, for alpha > 1 and almost all lambda is an element of (1/2, 1), the set E-lambda(alpha) = {x is an element of [0, 1] : vertical bar x - s(n vertical bar) < 2(-alpha n) infinitely often}, where s(n) is an element of {(1 - lambda)Sigma(n)(k=0) a(k)lambda(k) and a(k) is an element of {0, 1}}, belongs to the class G(s) for s <= 1/alpha. This improves one of our previously published results.

• Författare: Tomas Persson; Henry W. J. Reeve
• Avdelning/ar: Matematik LTH; Dynamical systems
• Publiceringsår: 2015
• Språk: Engelska
• Sidor: 521-542
• Publikation/Tidskrift/Serie: Proceedings of the Edinburgh Mathematical Society
• Volym: 58
• Issue: 2
• Länkar: https://arxiv.org/abs/1302.0954; http://dx.doi.org/10.1017/S0013091514000066
• Dokumenttyp: Artikel i tidskrift
• Förlag: Cambridge University Press
• Ämne: Mathematics
• Nyckelord: classes with large intersections; diophantine approximation; Hausdorff dimension
• Status: Published
• Forskningsgrupp: Dynamical systems
• ISBN/ISSN/Övrigt: ISSN: 1464-3839