Riesz $\alpha$-capacity of Cantor sets and cyclicity in Dirichlet-type spaces

We examine the threshold of the cyclicity for functions in Dirichlet-type spaces $\mathcal{D}_{\alpha}$, $\alpha\in(0,1]$. Given a fixed $\alpha^{*}\in(0,1]$, we construct a holomorphic function $f\in\mathcal{D}_{\alpha^{*}}$ which is cyclic in $\mathcal{D}_{\alpha}$ for all $\alpha<\alpha^{*}$, but fails to be cyclic in $\mathcal{D}_{\alpha^{*}}$. This function serves as a counterexample to the persistence of cyclicity at the critical index $\alpha^{*}$. Throughout the construction process, we work with generalized Cantor sets and study their Riesz $\alpha$-capacity.