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