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

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.