The additivity of the two-dimensional Miller ideal

Let J(M^2) denote the sigma-ideal associated with two-dimensional Miller forcing. We show that it is relatively consistent with ZFC that the additivity of J(M^2) is bigger than the covering number of the ideal of the meager subsets of the Baire space. We also show that Martin's Axiom implies that the additivity of J(M^2) is the cardinality of the continuum. Finally we prove that there are no analytical infinite maximal antichains in any finite product of the power set of the natural numbers modulo the ideal of the finite subsets of the natural numbers.