Not every 𝑄-set is perfectly meager in the transitive sense

Not every $Q$-set is perfectly meager in the transitive sense
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by Andrzej Nowik and Tomasz Weiss PDF

Proc. Amer. Math. Soc. 128 (2000), 3017-3024 Request permission

We prove the following theorems:

It is consistent with ZFC that there exists a $Q$ - set which is not perfectly meager in the transitive sense.
Every set which is perfectly meager in the transitive sense has the ${\overline {AFC}}$ property.
The product of two sets perfectly meager in the transitive sense has also that property.

References

Proceedings of the 12th winter school on abstract analysis, Circolo Matematico di Palermo, Palermo, 1984. Section of topology; Held at Srní, January 15–29, 1984; Rend. Circ. Mat. Palermo (2) 1984, Suppl. No. 6. MR 782700
E. Grzegorek, Always of the first category sets. II, Proceedings of the 13th winter school on abstract analysis (Srní, 1985), 1985, pp. 43–48 (1986). MR 894270
Haim Judah and Saharon Shelah, $Q$-sets, Sierpiński sets, and rapid filters, Proc. Amer. Math. Soc. 111 (1991), no. 3, 821–832. MR 1045594, DOI 10.1090/S0002-9939-1991-1045594-5
Arnold W. Miller, Special subsets of the real line, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 201–233. MR 776624
Andrej Nowik, Marion Scheepers, and Tomasz Weiss, The algebraic sum of sets of real numbers with strong measure zero sets, J. Symbolic Logic 63 (1998), no. 1, 301–324. MR 1610427, DOI 10.2307/2586602
Ireneusz Recław, Some additive properties of special sets of reals, Colloq. Math. 62 (1991), no. 2, 221–226. MR 1142923, DOI 10.4064/cm-62-2-221-226
J.v. Neumann, Ein System algebraisch unabhängiger Zahlen, Math. Ann. 99, 1928.
P. Zakrzewski, Universally meager sets, to appear.