A Frostman type lemma for sets with large intersections, and an application to Diophantine approximation

• Published in: arXiv.org
• Main Authors: Persson, Tomas, Reeve, Henry W J
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 11.09.2017
• Subjects: Intersections; Mathematics - Dynamical Systems; Mathematics - Number Theory
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1302.0954

We consider classes \(\mathscr{G}^s ([0,1])\) 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 \(E = \limsup E_n \subset [0,1]\), and that \(\mu_n\) are probability measures with support in \(E_n\). If there is a constant \(C\) such that \[\iint|x-y|^{-s}\, \mathrm{d}\mu_n(x)\mathrm{d}\mu_n(y)<C\] 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 \(\mathscr{G}^s ([0,1])\). As an application we prove that for \(\alpha > 1\) and almost all \(\lambda \in (\frac{1}{2},1)\) the set \[ E_\lambda(\alpha) = \{\,x\in[0,1] : |x - s_n| < 2^{-\alpha n} \text{infinitely often}\ \}\] where \(s_n \in \{\,(1-\lambda)\sum_{k=0}^na_k\lambda^k\) and \(a_k\in\{0,1\}\,\}\), belongs to the class \(\mathscr{G}^s\) for \(s \leq \frac{1}{\alpha}\). This improves one of our previous results.