Perfect tree forcings for singular cardinals

We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals 〈κ_n:n<ω〉, Prikry defined the forcing P of all perfect subtrees of ∏_n<ωκ_n, and proved that for κ=sup_n<ω⁡κ_n, assuming the necessary cardinal arithmetic, the Boolean completion B of P is (ω,μ)-distributive for all μ<κ but (ω,κ,δ)-distributivity fails for all δ<κ, implying failure of the (ω,κ)-d.l. These hitherto unpublished results are included, setting the stage for the following recent results. P satisfies a Sacks-type property, implying that B is (ω,∞,<κ)-distributive. The (h,2)-d.l. and the (d,∞,<κ)-d.l. fail in B. P(ω)/fin completely embeds into B. Also, B collapses κ^ω to h. We further prove that if κ is a limit of countably many measurable cardinals, then B adds a minimal degree of constructibility for new ω-sequences. Some of these results generalize to cardinals κ with uncountable cofinality.