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

TL;DR

This paper develops a Frostman type lemma for sets with large intersections, enabling the classification of certain limsup-sets within Falconer's classes, with applications to Diophantine approximation and Hausdorff dimension analysis.

Contribution

It introduces a Frostman type criterion for limsup-sets to belong to Falconer's classes, extending tools for analyzing large intersection properties.

Findings

Sets $E_ul ext{ in } $ belong to class ^s for $s \,\leq\, 1/\alpha$.

The criterion applies to sets defined by Diophantine approximation conditions.

Improves previous results on the intersection properties of these sets.

Abstract

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.