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Keywords:
meager-additive; $\mathcal E$-additive; strong measure zero; sharp measure zero; Hausdorff dimension; Hausdorff measure
Summary:
We develop a theory of sharp measure zero sets that parallels Borel's strong measure zero, and prove a theorem analogous to Galvin--Mycielski--Solovay theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $2^{\omega}$ is meager-additive if and only if it is $\mathcal E$-additive; if $f\colon 2^{\omega}\to 2^{\omega}$ is continuous and $X$ is meager-additive, then so is $f(X)$.
References:
[1] Bartoszyński T., Judah H.: Set Theory. On the structure of the real line. A K Peters, Wellesley, 1995. MR 1350295
[2] Bartoszyński T., Shelah S.: Closed measure zero sets. Ann. Pure Appl. Logic 58 (1992), no. 2, 93–110. DOI 10.1016/0168-0072(92)90001-G | MR 1186905
[3] Besicovitch A. S.: Concentrated and rarified sets of points. Acta Math. 62 (1933), no. 1, 289–300. DOI 10.1007/BF02393607 | MR 1555386
[4] Besicovitch A. S.: Correction. Acta Math. 62 (1933), no. 1, 317–318. DOI 10.1007/BF02393610 | MR 1555389
[5] Borel E.: Sur la classification des ensembles de mesure nulle. Bull. Soc. Math. France 47 (1919), 97–125 (French). DOI 10.24033/bsmf.996 | MR 1504785
[6] Carlson T. J.: Strong measure zero and strongly meager sets. Proc. Amer. Math. Soc. 118 (1993), no. 2, 577–586. DOI 10.1090/S0002-9939-1993-1139474-6 | MR 1139474
[7] Corazza P.: The generalized Borel conjecture and strongly proper orders. Trans. Amer. Math. Soc. 316 (1989), no. 1, 115–140. DOI 10.1090/S0002-9947-1989-0982239-X | MR 0982239
[8] Federer H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, 153, Springer, New York, 1969. MR 0257325 | Zbl 0874.49001
[9] Fremlin D. H.: Measure Theory. Vol. 5, Set-theoretic Measure Theory, Part I. Torres Fremlin, Colchester, 2015. MR 3723041
[10] Galvin F., Miller A. W.: {$\gamma $. -sets and other singular sets of real numbers}, Topology Appl. 17 (1984), no. 2, 145–155. DOI 10.1016/0166-8641(84)90038-5 | MR 0738943
[11] Galvin F., Mycielski J., Solovay R. M.: Strong measure zero sets. Abstract 79T–E25, Not. Am. Math. Soc. 26 (1979), A-280.
[12] Galvin F., Mycielski J., Solovay R. M.: Strong measure zero and infinite games. Arch. Math. Logic 56 (2017), no. 7–8, 725–732. DOI 10.1007/s00153-017-0541-z | MR 3696064
[13] Gerlits J., Nagy Z.: Some properties of {$C(X)$. I}, Topology Appl. 14 (1982), no. 2, 151–161. DOI 10.1016/0166-8641(82)90065-7 | MR 0667661
[14] Gödel K.: The consistency of the axiom of choice and of the generalized continuum-hypothesis. Proc. Natl. Acad. Sci. USA 24 (1938), no. 12, 556–557. DOI 10.1073/pnas.24.12.556
[15] Gödel K.: The Consistency of the Continuum Hypothesis. Annals of Mathematics Studies, 3, Princeton University Press, Princeton, 1940. MR 0002514
[16] Howroyd J. D.: On the Theory of Hausdorff Measures in Metric Spaces. Ph.D. Thesis, University College, London, 1994. MR 1365084
[17] Howroyd J. D.: On Hausdorff and packing dimension of product spaces. Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 4, 715–727. DOI 10.1017/S0305004100074545 | MR 1362951
[18] Hrušák M., Wohofsky W., Zindulka O.: Strong measure zero in separable metric spaces and Polish groups. Arch. Math. Logic 55 (2016), no. 1–2, 105–131. DOI 10.1007/s00153-015-0459-2 | MR 3453581
[19] Hrušák M., Zapletal J.: Strong measure zero sets in Polish groups. Illinois J. Math. 60 (2016), no. 3–4, 751–760. DOI 10.1215/ijm/1506067289 | MR 3707641
[20] Kelly J. D.: A method for constructing measures appropriate for the study of Cartesian products. Proc. London Math. Soc. (3) 26 (1973), 521–546. MR 0318427
[21] Kysiak M.: On Erdős-Sierpiński Duality between Lebesgue Measure and Baire Category. Master's Thesis, Uniwersytet Warszawski, Warszawa, 2000 (Polish).
[22] Laver R.: On the consistency of Borel's conjecture. Acta Math. 137 (1976), no. 3–4, 151–169. DOI 10.1007/BF02392416 | MR 0422027 | Zbl 0357.28003
[23] Munroe M. E.: Introduction to Measure and Integration. Addison-Wesley Publishing Company, Cambridge, 1953. MR 0053186
[24] Nowik A., Scheepers M., Weiss T.: The algebraic sum of sets of real numbers with strong measure zero sets. J. Symbolic Logic 63 (1998), no. 1, 301–324. DOI 10.2307/2586602 | MR 1610427 | Zbl 0901.03036
[25] Nowik A., Weiss T.: On the Ramseyan properties of some special subsets of $2^\omega$ and their algebraic sums. J. Symbolic Logic 67 (2002), no. 2, 547–556. DOI 10.2178/jsl/1190150097 | MR 1905154
[26] Pawlikowski J.: A characterization of strong measure zero sets. Israel J. Math. 93 (1996), 171–183. DOI 10.1007/BF02761100 | MR 1380640 | Zbl 0857.28001
[27] Rogers C. A.: Hausdorff Measures. Cambridge University Press, London, 1970. MR 0281862
[28] Scheepers M.: Finite powers of strong measure zero sets. J. Symbolic Logic 64 (1999), no. 3, 1295–1306. DOI 10.2307/2586631 | MR 1779763
[29] Shelah S.: Every null-additive set is meager-additive. Israel J. Math. 89 (1995), no. 1–3, 357–376. DOI 10.1007/BF02808209 | MR 1324470
[30] Sierpiński W.: Sur un ensemble non denombrable, dont toute image continue est de mesure nulle. Fundamenta Mathematicae 11 (1928), no. 1, 302–304 (French). DOI 10.4064/fm-11-1-302-303
[31] Tsaban B., Weiss T.: Products of special sets of real numbers. Real Anal. Exchange 30 (2004/05), no. 2, 819–835. DOI 10.14321/realanalexch.30.2.0819 | MR 2177439
[32] Weiss T.: On meager additive and null additive sets in the Cantor space $2^\omega$ and in $\Bbb R$. Bull. Pol. Acad. Sci. Math. 57 (2009), no. 2, 91–99. MR 2545840
[33] Weiss T.: Addendum to “On meager additive and null additive sets in the Cantor space $2^\omega$ and in $\Bbb R$” (Bull. Polish Acad. Sci. Math. 57 (2009), 91–99). Bull. Pol. Acad. Sci. Math. 62 (2014), no. 1, 1–9. DOI 10.4064/ba57-2-1 | MR 3241126
[34] Weiss T.: Properties of the intersection ideal $\mathcal M\cap \mathcal N$ revisited. Bull. Pol. Acad. Sci. Math. 65 (2017), no. 2, 107–111. DOI 10.4064/ba8098-8-2017 | MR 3731016
[35] Weiss T., Tsaban B.: Topological diagonalizations and Hausdorff dimension. Note Mat. 22 (2003/04), no. 2, 83–92. MR 2112732
[36] Wohofsky W.: Special Sets of Real Numbers and Variants of the Borel Conjecture. Ph.D. Thesis, Technische Universität Wien, Wien, 2013.
[37] Zakrzewski P.: Universally meager sets. Proc. Amer. Math. Soc. 129 (2001), no. 6, 1793–1798. DOI 10.1090/S0002-9939-00-05726-9 | MR 1814112
[38] Zakrzewski P.: Universally meager sets. II. Topology Appl. 155 (2008), no. 13, 1445–1449. DOI 10.1016/j.topol.2008.05.005 | MR 2427418
[39] Zindulka O.: Universal measure zero, large Hausdorff dimension, and nearly Lipschitz maps. Fund. Math. 218 (2012), no. 2, 95–119. DOI 10.4064/fm218-2-1 | MR 2957686
[40] Zindulka O.: Packing measures and dimensions on Cartesian products. Publ. Mat. 57 (2013), no. 2, 393–420. DOI 10.5565/PUBLMAT_57213_06 | MR 3114775
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