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Analytic Hausdorff gaps II: The density zero ideal

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Abstract

We prove two results about the quotient over the asymptotic density zero ideal. First, it is forcing equivalent to\(\mathcal{P}\left( \mathbb{N} \right)/Fin * \mathcal{R}_c \)% MathType!End!2!1!, where\(\mathcal{P}\left( \mathbb{N} \right)/\mathcal{I}\)% MathType!End!2!1! is the homogeneous probability measure algebra of characterc. Second, if it has analytic Hausdorff gaps, then they look considerably different from proviously known gaps of this form.

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References

  1. B. Balcar, F. Hernández-Hernández and M. Hrušák,Combinatorics of dense subsets of the rationals, Forum Mathematicum183 (2004), 59–80.

    MATH  Google Scholar 

  2. M. Benedikt,Ultrafilters which extend measures, The Journal of Symbolic Logic63 (1998), 638–662.

    Article  MATH  MathSciNet  Google Scholar 

  3. I. Farah,Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Memoirs of the American Mathematical Society148 (2000), no. 702, 177 pp.

  4. I. Farah,Analytic Hausdorff gaps, inSet Theory, DIMACS Series, Vol. 58, American Mathematical Society, Providence, RI, 2002, pp. 65–72.

    Google Scholar 

  5. I. Farah, How many Boolean algebras\(\mathcal{P}\left( \mathbb{N} \right)/\mathcal{I}\)% MathType!End!2!1! are there?, Illinois Journal of Mathematics46 (2003), 999–1033.

    MathSciNet  Google Scholar 

  6. I. Farah,Luzin gaps, Transactions of the American Mathematical Society356 (2004), 2197–2239.

    Article  MATH  MathSciNet  Google Scholar 

  7. I. Farah,Rigidity conjectures, inProceedings of Logic Colloquium 2000 (R. Cori, A. Razborov, S. Todorcevic and C. Wood, eds.), Association for Symbolic Logic, Poughkeepsie, New York, 2005, pp. 252–271.

    Google Scholar 

  8. I. Farah and J. Zapletal,Four andmore, Annals of Pure and Applied Logic, to appear.

  9. D. H. Fremlin,Measure Theory, Vol. 3, Torres-Fremlin, Colchester, UK, 2002.

    Google Scholar 

  10. D. H. Fremlin,Measure Theory, Vol. 4, Torres-Fremlin, Colchester, UK, 2003.

    Google Scholar 

  11. C. W. Henson and B. Wattenberg,Egoroff's theorem and the distribution of standard points in a nonstandard model, Proceedings of the American Mathematical Society81 (1981), 455–461.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Hruŝák and J. Zapletal,Forcing with quotients, preprint, 2004.

  13. W. Just,A weak version of AT from OCA, Mathematical Sciences Research Institute Publications26 (1992), 281–291.

    MathSciNet  Google Scholar 

  14. W. Just and A. Krawczyk, On certain Boolean algebras\(\mathcal{P}\left( \omega \right)/\mathcal{I}\)% MathType!End!2!1!, Transactions of the American Mathematical Society285 (1984), 411–429.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. R. D. Mathias,Happy families, Annals of Mathematical Logic12 (1977), 59–111.

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Solecki,Analytic ideals and their applications, Annals of Pure and Applied Logic99 (1999), 51–72.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Steprāns,Many quotient algebras of the integers modulo co-analytic ideals, preprint, York University, 2002, available at http://www.math.yorku/ca/~steprans/.

  18. S. Todorcevic,Partition Problems in Topology, Contemporary Mathematics, Vol. 84, American Mathematical Society, Providence, RI, 1989.

    MATH  Google Scholar 

  19. S. Todorcevic,Analytic gaps, Fundamenta Mathematicae150 (1996), 55–67.

    MATH  MathSciNet  Google Scholar 

  20. S. Todorcevic,Gaps in analytic quotients, Fundamental Mathematicae156 (1998), 85–97.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, 4700 Keele Street, M3J 1P3, Toronto, ON, Canada

    Ilijas Farah

  2. Matematicki Institut, Kneza Mihaila 35, Belgrade, Serbia and Montenegro

    Ilijas Farah

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  1. Ilijas Farah
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Partially supported by NSERC.

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Farah, I. Analytic Hausdorff gaps II: The density zero ideal. Isr. J. Math. 154, 235–246 (2006). https://doi.org/10.1007/BF02773608

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  • DOI: https://doi.org/10.1007/BF02773608

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