A classification of trigonometrical thin sets and their interrelations

Elias, Peter

doi:10.1090/S0002-9939-97-03661-7

A classification of trigonometrical thin sets and their interrelations
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Proc. Amer. Math. Soc. 125 (1997), 1111-1121 Request permission

Abstract:

We introduce a uniform way of classifying thin sets of harmonic analysis related to absolute convergence of trigonometric series. This classification covers classical classes $(\mathcal {D},\mathcal {P}\mathcal {D},\mathcal {A}, \mathcal {N}_0,\mathcal {N})$ and yields two new ones ($\mathcal {B}_0$ and $\mathcal {B})$. We study interrelation between these classes concerning combinatorial structure of thin sets.

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, Combinatorial aspects of measure and category, Fund. Math. 127 (1987), no. 3, 225–239. MR 917147, DOI 10.4064/fm-127-3-225-239
N. K. Bary, A treatise on trigonometric series. Vols. I, II, A Pergamon Press Book, The Macmillan Company, New York, 1964. Authorized translation by Margaret F. Mullins. MR 0171116
L. Bukovský, N. N. Kholshchevnikova and M. Repický, Thin sets of harmonic analysis and infinite combinatorics, to appear in Real Anal. Exchange.
Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
P. Eliaš, Systemization of classes of thin sets related to the absolute convergence of trigonometric series (in Slovak), Master thesis, Univerzita P. J. Šafárika, Košice, 1993.
Sylvain Kahane, Antistable classes of thin sets in harmonic analysis, Illinois J. Math. 37 (1993), no. 2, 186–223. MR 1208819
J. Marcinkiewicz, Quelques théorèmes sur les séries et les fonctions, Bull. Sém. Math. Univ. Wilno 1 (1938), 19–24.
Arnold W. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), no. 1, 93–114. MR 613787, DOI 10.1090/S0002-9947-1981-0613787-2
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
Jan van Mill and George M. Reed (eds.), Open problems in topology, North-Holland Publishing Co., Amsterdam, 1990. MR 1078636
T. Viola, Sull’insieme dei punti di convergenza delle serie trigonometriche generali, Ann. Scuola Norm. Sup. Pisa (2) 4 (1935), 155–162.