@TECHREPORT{Farah04analytichausdorff, author = {Ilijas Farah}, title = {Analytic Hausdorff gaps II: Density zero ideal}, institution = {}, year = {2004} }

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Abstract

Abstract. 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 sup n 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