A lacunary version of Mergelian's approximation theorem

Let K be a compact plane set having connected complement. Then Mergelian's theorem states that the linear span of the monomials zn, i.e. the polynomials, are dense in the Banach space A(K) of all functions continuous on K and holomorphic in the interior of K endowed with the sup-norm. We consider the question under which conditions the linear span of zn, with n running through a sequence of nonnegative integers having upper density one, is dense in A(K) or appropriate subspaces.