A family of permitted trigonometric thin sets

Repický, Miroslav

doi:10.1090/S0002-9939-97-03516-8

A family of permitted trigonometric thin sets
HTML articles powered by AMS MathViewer

by Miroslav Repický PDF

Proc. Amer. Math. Soc. 125 (1997), 137-144 Request permission

Abstract:

We introduce the notion of perfectly measure zero sets and prove that every perfectly measure zero set is permitted for the families of all pseudo-Dirichlet sets, N$_{0}$-sets, A-sets and N-sets. In particular this means that these families of trigonometric thin sets are closed under adding sets of cardinality less than the additivity of Lebesgue measure.

References

P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Tomek Bartoszyński, Additivity of measure implies additivity of category, Trans. Amer. Math. Soc. 281 (1984), no. 1, 209–213. MR 719666, DOI 10.1090/S0002-9947-1984-0719666-7
Bartoszyński T. and Recław I., Not every $\gamma$-set is strongly meager, preprint.
Bartoszyński T. and Scheepers M., Remarks on sets related to trigonometric series, Topology Appl. 64 (1995), 133-140.
Bary N. K., A Treatise on Trigonometric Series, Macmillan, New York, 1964.
Zuzana Bukovská and Lev Bukovský, Adding small sets to an $\textbf {N}$-set, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3867–3873. MR 1285977, DOI 10.1090/S0002-9939-1995-1285977-5
Bukovský L., Kholshchevnikova N. N. and Repický M., Thin sets of harmonic analysis and infinite combinatorics, Real Analysis Exchange 20 (1994/95), 454–509.
Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
J. Gerlits and Zs. Nagy, Some properties of $C(X)$. I, Topology Appl. 14 (1982), no. 2, 151–161. MR 667661, DOI 10.1016/0166-8641(82)90065-7
Martin Goldstern, Tools for your forcing construction, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 305–360. MR 1234283
N. N. Kholshchevnikova, Uncountable $R$- and $N$-sets, Mat. Zametki 38 (1985), no. 2, 270–277, 349 (Russian). MR 808896
Recław I., Private communication.
Miroslav Repický, Goldstern-Judah-Shelah preservation theorem for countable support iterations, Fund. Math. 144 (1994), no. 1, 55–72. MR 1271478, DOI 10.4064/fm-144-1-55-72