## More remarks on the intersection ideal $$\mathcal{M}\cap\mathcal{N}$$.(English)Zbl 1424.03020

Given two ideal $$I$$ and $$J$$ on the Cantor set $$2^\omega$$ we say that $$X\in(I,J)^*$$ if for all $$A$$ in $$I$$ the sum $$A+X$$ is in $$J$$; also, $$I^*$$ abbreviates $$(I,I)^*$$. Answering a question from [the author, Commentat. Math. Univ. Carol. 54, No. 3, 437–445 (2013; Zbl 1289.03031)], the author shows that $$(\mathcal{M}\cap \mathcal{N})^*\subseteq \mathcal{N}^*$$, where $$\mathcal{M}$$ and $$\mathcal{N}$$ are the ideals of meager and null sets, respectively. In addition, there is a provisional solution to the question whether it is consistent that every member of $$(\mathcal{E},\mathcal{M})^*$$ is countable, where $$\mathcal{E}$$ is the $$\sigma$$-ideal generated by the $$F_\sigma$$-sets of measure zero.
Reviewer: K. P. Hart (Delft)

### MSC:

 300000 Other combinatorial set theory 3e+17 Cardinal characteristics of the continuum 3e+15 Descriptive set theory

### Keywords:

 [1] Bartoszyński T., Remarks on small sets of reals, Proc. Amer. Math. Soc 131 (2003), no. 2, 625-630 · Zbl 1017.03027 [2] Bartoszyński T.; Judah H., Set Theory. On the Structure of the Real Line, A K Peters, Wellesley, 1995 · Zbl 0834.04001 [3] Goldstern M.; Kellner J.; Shelah S.; Wohofsky W., Borel conjecture and dual Borel conjecture, Trans. Amer. Math. Soc. 366 (2014), no. 1, 245-307 · Zbl 1298.03106 [4] Orenshtein T.; Tsaban B., Linear $$σ$$-additivity and some applications, Trans. Amer. Math. Soc. 363 (2011), no. 7, 3621-3637 · Zbl 1223.54032 [5] Pawlikowski J., A characterization of strong measure zero sets, Israel J. Math. 93 (1996), 171-183 · Zbl 0857.28001 [6] Weiss T., A note on the intersection ideal $${\mathcal{M}}∩ {\mathcal{N}}$$, Comment. Math. Univ. Carolin. 54 (2013), no. 3, 437-445