Critical cyclicity in Dirichlet-type spaces on the bidisk

Consider the Dirichlet-type spaces on the bidisc defined by $${\mathcal{D}}_{\beta }({\mathbb{D}}^2)=\{f(z_1,z_2)=\sum _{k,l}{a}_{kl}{z}_1^k{z}_2^l\in \mathcal{O}({\mathbb{D}}^2):\sum _{k,l\ge 0}\mathrm{\mid }{a}_{kl}{\mathrm{\mid }}^2(k+l+1{)}^{\beta }<+\mathrm{\infty }\}\mathrm{.}$$ Given ${\beta }_c\in (0,2],$ we construct a function $f$ that belongs to the Dirichlet-type space ${\mathcal{D}}_2({\mathbb{D}}^2)$ of the bidisk and is cyclic in ${\mathcal{D}}_{\beta }({\mathbb{D}}^2)$ if and only if $\beta \le {\beta }_c\mathrm{.}$ We also show that the critical index satisfies ${\beta }_c=2-{\mathrm{d}\mathrm{i}\mathrm{m}}_H(\mathcal{Z}(f)\cap {\mathbb{T}}^2),$ where $\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{H}}(\mathcal{Z}(f)\cap {\mathbb{T}}^2)$ is the Hausdorff dimension of the zero set of the function $f$ on the two-torus ${\mathbb{T}}^2\mathrm{.}$