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Adding small sets to an $ {\bf N}$-set


Authors: Zuzana Bukovská and Lev Bukovský
Journal: Proc. Amer. Math. Soc. 123 (1995), 3867-3873
MSC: Primary 04A15; Secondary 03E05, 03E15, 03E50, 04A20, 42A99, 43A46
DOI: https://doi.org/10.1090/S0002-9939-1995-1285977-5
MathSciNet review: 1285977
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Abstract: Pseudo Dirichlet and N-sets are small sets of reals defined in the theory of trigonometric series. We prove that by adding a set of cardinality smaller than $ \mathfrak{p}$ to an N-set one obtains again an N-set. This is a strengthening of Arbault-Erdös' theorem about adding countable sets to N-sets. A similar result holds true for pseudo Dirichlet sets.


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  • [Ar] J. Arbault, Sur l'ensemble de convergence absolue d'une série trigonometrique, Bull. Soc. Math. France 80 (1952), 253-317. MR 0055476 (14:1080d)
  • [Ba] N. K. Bary, A treatise on trigonometrical series, GIFML, Moskva, 1961; English transl., Pergamon Press, Oxford, 1965.
  • [Be] M. G. Bell, On the combinatorial principle $ P(\mathfrak{c})$, Fund. Math. 114 (1981), 149-157. MR 643555 (83e:03077)
  • [B1] Z. Bukovská, Thin sets in trigonometrical series and quasinormal convergence, Math. Slovaca 40 (1990), 53-62. MR 1094972 (92b:43010)
  • [B2] -, Quasinormal convergence, Math. Slovaca 41 (1991), 137-146. MR 1108577 (92k:26015)
  • [BL] L. Bukovský, Thin sets related to trigonometric series, Set Theory of the Reals, Israel Math. Conf. Proc. (H. Judah, ed.), vol. 6, Bar-Ilan University, Ramat-Gan, 1993, pp. 107-118. MR 1234280 (94h:42021)
  • [Ca] Cassels, An introduction to Diophantine approximation, Cambridge Univ. Press, Cambridge, 1965.
  • [CL] Á. Császár and M. Laczkovich, Discrete and equal convergence, Studia Sci. Math. Hungar. 10 (1975), 463-472. MR 515347 (81e:54013)
  • [vD] E. K. van Douwen, The integers and topology, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. 111-167. MR 776619 (85k:54001)
  • [Fr] D. Fremlin, Consequences of Martin's Axiom, Cambridge Univ. Press, Cambridge, 1984. MR 780933 (86i:03001)
  • [Je] T. Jech, Set theory, Academic Press, New York, 1978. MR 506523 (80a:03062)
  • [KJ] J. P. Kahane, Séries de Fourier absolument convergentes, Springer-Verlag, Berlin, 1970. MR 0275043 (43:801)
  • [KS] S. Kahane, Antistable classes of thin sets in harmonic analysis, Illinois J. Math. 37 (1993), 186-223. MR 1208819 (94g:43005)
  • [Kh] N. N. Kholshchevnikova, O neshchetnykh R- i N-mnozhestvakh, Mat. Zametki 38 (1985), 270-277. MR 808896 (87b:03115)
  • [Li] L.-A. Lindahl, Dirichlet, Kronecker and Helson sets, Thin Sets in Harmonic Analysis (L.-A. Lindahl and F. Poulsen, eds.), Marcel Dekker, New York, 1971, pp. 1-19. MR 0440283 (55:13161)
  • [Sa] R. Salem, On some properties of symmetrical perfect sets, Bull. Amer. Math. Soc. 47 (1941), 820-82. MR 0005132 (3:105f)
  • [Vo] P. Vojtáš, Boolean isomorphism between partial orderings of convergent and divergent series and infinite subsets of $ \mathbb{N}$, Proc. Amer. Math. Soc. 117 (1993), 235-242. MR 1106183 (93c:03068)
  • [Z] A. Zygmund, Trigonometric series I, Cambridge Univ. Press, Cambridge, 1959. MR 0107776 (21:6498)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1285977-5
Keywords: Trigonometric series, N-set, pseudo Dirichlet set, $ \mathfrak{p}$
Article copyright: © Copyright 1995 American Mathematical Society

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