Logarithmic capacity of random $G_\delta$-sets

We study the logarithmic capacity of $G_\delta$ subsets of the interval $[0,1].$ Let $S$ be of the form \begin{align*} S=\bigcap_m \bigcup_{k\ge m} I_k, \end{align*} where each $I_k$ is an interval in $[0,1]$ with length $l_k$ that decrease to $0$. We provide sufficient conditions for $S$ to have full capacity, i.e. $\mathop{\mathrm{Cap}}(S)=\mathop{\mathrm{Cap}}([0,1])$. We consider the case when the intervals decay exponentially and are placed in $[0,1]$ randomly with respect to some given distribution. The random $G_\delta$ sets generated by such distribution satisfy our sufficient conditions almost surely and hence, have full capacity almost surely. This study is motivated by the $G_\delta$ set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products. We also study the family of $G_\delta$ sets $\{S(\alpha)\}_{\alpha>0}$ that are generated by setting the decreasing speed of the intervals to $l_k=e^{-k^\alpha}.$ We observe a sharp transition from full capacity to zero capacity by varying $\alpha>0$.