More on Cardinal Invariants of Analytic P-ideals

Given an ideal I on ω let a(I) (ā(I)) be minimum of the cardinalities of in nite (uncountable) maximal I-almost disjoint subsets of [ω]. We show that a(Ih) > ω if Ih is a summable ideal; but a(Z~ μ) = ω for any tall density ideal Z~ μ including the density zero ideal Z. On the other hand, you have b ≤ ā(I) for any analytic P -ideal I, and ā(Z~ μ) ≤ a for each density ideal Z~ μ. For each ideal I on ω denote bI and dI the unbounding and dominating numbers of 〈ω,≤I〉 where f ≤I g i {n ∈ ω : f(n) > g(n)} ∈ I. We show that bI = b and dI = d for each analytic P-ideal I. Given a Borel ideal I on ω we say that a poset P is I-bounding i ∀x ∈ I∩V P ∃y ∈ I∩V x ⊆ y. P is I-dominating i ∃y ∈ I∩V P ∀x ∈ I ∩ V x ⊆∗ y. For each analytic P-ideal I if a poset P has the Sacks property then P is I-bounding; moreover if I is tall as well then the property I-bounding/I-dominating implies ω-bounding/adding dominating reals, and the converses of these two implications are false. For the density zero ideal Z we can prove more: (i) a poset P is Z-bounding i it has the Sacks property, (ii) if P adds a slalom capturing all ground model reals then P is Z-dominating.