Analytic Hausdorff Gaps Ii: the Density Zero Ideal

We prove two results about the quotient over the asymptotic density zero ideal. First, it is forcing equivalent to P(N)/Fin ∗Rc, where Rc is the homogeneous probability measure algebra of character c. Second, if it has analytic Hausdorff gaps then they look considerably different from previously known gaps of this form. We consider density ideals, ideals of the form Zμ = {A| lim supn μn(A) = 0} for a sequence μm (m ∈ N) of probability measures concentrating on pairwise disjoint intervals Im (m ∈ N) of N. In Theorem 1.3 we prove that the regular open algebra of such quotient is isomorphic to the regular open algebra of P(N)/Fin ∗Rc. Study of quotients P(N)/I as forcing notions has recently attracted a bit of attention ([1], [12], [8]). In [19] it was proved that there are no analytic Hausdorff gaps over Fin. Todorcevic actually proved that every pregap A, B over Fin such that A is analytic and B/Fin is σ-directed can be countably separated (and more). In [3, Theorem 5.7.1, Theorem 5.7.2 and Lemma 5.8.7] we have proved that Fin is the only analytic P-ideal that has this property: If I is an analytic P-ideal that is not Rudin–Keisler isomorphic to Fin, then there is a gap A, B over I such that A and B are Borel, B/I is σ-directed and A is not countably separated from B. In [4] it was proved that there are analytic Hausdorff gaps over any dense Fσ P-ideal. Recall that Z0 = {A ⊆ N : lim sup n |A ∩ n|/n = 0} is the ideal of asymptotic density zero sets. In §2 we prove results on the structure of analytic Hausdorff gaps in its quotients, making some progress towards [3, Question 5.13.7] and [4, Question 8a and Question 10]. In Proposition 3.2 we show that if I is a dense analytic P-ideal without analytic Hausdorff gaps in its quotient, then the restriction of I to some positive set is a generalized density ideal. This gives a partial solution to the problem of characterizing those analytic P-ideals that do not have analytic Hausdorff gaps in their quotients ([3, Problem 5.13.5]; see also Question 4.1). Terminology. Our terminology and notation follow [3]. Two families A,B in a quotient P(N)/I form a pregap if A ∩ B ∈ I for all A ∈ A and B ∈ B. A pregap is separated (or split) by C ⊆ N if for for every A ∈ A and B ∈ B we have A \ C ∈ I and B ∩ C ∈ I. If it is not separated by any C, then it is a gap. Date: February 16, 2005. 1991 Mathematics Subject Classification. 03E15, 03E05, 03C20, 06E05. Filename: n2003d08-s.tex. Partially supported by NSERC.