On the Brown--Shields conjecture for cyclicity in the Dirichlet space

Let $\text{cD}$ be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We establish a new sufficient condition for a function $f\in \text{cD}$ to be {\em cyclic}, i.e. for $\{pf:p\text{a polynomial}\}$ to be dense in $\text{cD}$. This allows us to prove a special case of the conjecture of Brown and Shields that a function is cyclic in $\text{cD}$ iff it is outer and its zero set (defined appropriately) is of capacity zero.