Cardinal characteristics associated with small subsets of reals
Abstract: Inspired by Bartoszy\'nski's work on small sets, we introduce a new ideal defined by interval partitions on natural numbers and summable sequences of positive reals. Similarly, we present another ideal that relies on Bartoszy\'nski's and Shelah's representation of $F_\sigma$ measure zero sets. We show they are $\sigma$-ideals characterizing all small sets and $F_\sigma$ measure zero sets. We also study the cardinal characteristics associated with the introduced ideals. We use them to describe the invariants of measure, discuss their connection to Cicho\'n's diagram, and present related consistency results.
- Tomek Bartoszynski. Additivity of measure implies additivity of category. Transactions of the American Mathematical Society, 281(1):209–213, 1984.
- Tomek Bartoszynski. On covering of real line by null sets. Pacific Journal of Mathematics, 131(1):1–12, 1988.
- Filter-linkedness and its effect on preservation of cardinal characteristics. Ann. Pure Appl. Logic, 172(1):Paper No. 102856, 30, 2021.
- Separating cardinal characteristics of the strong measure zero ideal. Preprint, arXiv:2309.01931, 2023.
- Set Theory: On the Structure of the Real Line. Ak Peters Series. Taylor & Francis, 1995.
- Andreas Blass. Combinatorial cardinal characteristics of the continuum. In Handbook of set theory. Vols. 1, 2, 3, pages 395–489. Springer, Dordrecht, 2010.
- Rothberger gaps in fragmented ideals. Fund. Math., 227(1):35–68, 2014.
- Jörg Brendle. Between P𝑃Pitalic_P-points and nowhere dense ultrafilters. Isr. J. Math., 113:205–230, 1999.
- Jörg Brendle. Forcing and the structure of the real line. Lecture notes for the mini-course of the same name at the Universidad Nacional of Colombia, 2009.
- Closed measure zero sets. Ann. Pure Appl. Logic, 58(2):93–110, 1992.
- A note on small sets of reals. C. R. Math., 356(11):1053–1061, 2018.
- Miguel A. Cardona. A friendly iteration forcing that the four cardinal characteristics of ℰℰ\mathcal{E}caligraphic_E can be pairwise different. Colloq. Math., 173(1):123–157, 2023.
- Miguel A. Cardona. The cardinal characteristics of the ideal generated by the Fσ measure zero subsets of the reals. Kyōto Daigaku Sūrikaiseki Kenkyūsho Kōkyūroku, 2024. To appear, arXiv:2402.04984.
- On cardinal characteristics of Yorioka ideals. MLQ, 65(2):170–199, 2019.
- Localization and anti-localization cardinals. arXiv:2305.03248, 2023.
- Uniformity numbers of the null-additive and meager-additive ideals. Preprint, arXiv:2401.15364, 2024.
- The left side of Cichoń’s diagram. Proc. Amer. Math. Soc., 144(9):4025–4042, 2016.
- Diego Alejandro Mejía. Matrix iterations and Cichoń’s diagram. Arch. Math. Logic, 52(3-4):261–278, 2013.
- Diego A. Mejía. Matrix iterations with vertical support restrictions. In Proceedings of the 14th and 15th Asian Logic Conferences, pages 213–248. World Sci. Publ., Hackensack, NJ, 2019.
- Saharon Shelah. Every null-additive set is meager-additive. Isr. J. Math., 89:357– 376, 1995.
- Saharon Shelah. Covering of the null ideal may have countable cofinality. Fund. Math., 166(1-2):109–136, 2000.
- Peter Vojtáš. Generalized Galois-Tukey-connections between explicit relations on classical objects of real analysis. In Set theory of the reals (Ramat Gan, 1991), volume 6 of Israel Math. Conf. Proc., pages 619–643. Bar-Ilan Univ., Ramat Gan, 1993.
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