Cyclicity in the Dirichlet space

Let D be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We give a new sufficient condition, not far from the known necessary condition, for a function f is an element of D to be cyclic, i.e. for {pf:p is a polynomial} to be dense in D. The proof is based on the notion of Bergman-Smirnov exceptional set introduced by Hedenmalm and Shields. Our methods yield the first known examples of such sets that are uncountable. One of the principal ingredients of the proof is a new converse to the strong-type inequality for capacity.