Approximation of polynomial hulls by analytic varieties: A counterexample

We construct a connected, compact set K ⊂ C² with the following property: there exist points p ∈ K̂ K such that there does not exist a sequence _ν\ of analytic sets A_ν⊂⊂ C² with boundary satisfying p ∈ A_ν for every ν∈ N and limν→∞ bA_ν⊂ K. For every point in K̂ K, we explicitly construct a sequence of Poletsky discs, and we compute the weak limit of the pushforwards of the Green current under these discs.