Cyclicity in Dirichlet-type spaces on the bidisk

Abstract: Consider the Dirichlet-type spaces on the bidisc defined by $$\mathcal{D}<em>β(\mathbb{D}^{2)=\Bigg{f(z_1,z_2)=\sum}</em>{k,l}a_{kl}z_1^{kz_2^{l\in\mathcal{O}(\mathbb{D}^2):}} \sum_{k,l\ge 0}|a_{kl}|<sup>2(k+l+1)<sup>β&lt;+\infty\Bigg}.$$ Given $β<em>c\in(0,2],$ we construct a function $f$ that belongs to the Dirichlet-type space $\mathcal D</em>{2}(\mathbb{D}<sup>2)$ of the bidisk and is cyclic in $\mathcal D_β(\mathbb{D}<sup>2)$ if and only if $β\leq β_{c}.$ We also show that the critical index satisfies $$β_c=2-\mathrm{dim}_H(\mathcal{Z}(f)\cap \mathbb{T}<sup>2),$$ where $$\mathrm{dim_H}(\mathcal{Z}(f)\cap \mathbb{T}<sup>2)$$ is the Hausdorff dimension of the zero set of the function $f$ on the two-torus $\mathbb{T}<sup>2.$