Finite union of H-sets and countable compact sets

Colloquium Mathematicae (1993)

• Volume: 65, Issue: 1, page 83-86
• ISSN: 0010-1354

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Abstract

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In [2], D. E. Grow and M. Insall construct a countable compact set which is not the union of two H-sets. We make precise this result in two directions, proving such a set may be, but need not be, a finite union of H-sets. Descriptive set theory tools like Cantor-Bendixson ranks are used; they are developed in the book of A. S. Kechris and A. Louveau [6]. Two proofs are presented; the first one is elementary while the second one is more general and useful. Using the last one I prove in my thesis, directed by A. Louveau, the existence of a countable compact set which is not a finite union of Dirichlet sets. This result, quoted in [3], is weaker because all Dirichlet sets belong to H. Other new results about the class H and similar classes of thin sets can be found in [4], [1] and [5].

How to cite

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Kahane, Sylvain. "Finite union of H-sets and countable compact sets." Colloquium Mathematicae 65.1 (1993): 83-86. <http://eudml.org/doc/210208>.

@article{Kahane1993,
abstract = {In [2], D. E. Grow and M. Insall construct a countable compact set which is not the union of two H-sets. We make precise this result in two directions, proving such a set may be, but need not be, a finite union of H-sets. Descriptive set theory tools like Cantor-Bendixson ranks are used; they are developed in the book of A. S. Kechris and A. Louveau [6]. Two proofs are presented; the first one is elementary while the second one is more general and useful. Using the last one I prove in my thesis, directed by A. Louveau, the existence of a countable compact set which is not a finite union of Dirichlet sets. This result, quoted in [3], is weaker because all Dirichlet sets belong to H. Other new results about the class H and similar classes of thin sets can be found in [4], [1] and [5].},
author = {Kahane, Sylvain},
journal = {Colloquium Mathematicae},
keywords = {-sets; countable compact set; Cantor-Bendixson ranks; Dirichlet sets},
language = {eng},
number = {1},
pages = {83-86},
title = {Finite union of H-sets and countable compact sets},
url = {http://eudml.org/doc/210208},
volume = {65},
year = {1993},
}

TY - JOUR
AU - Kahane, Sylvain
TI - Finite union of H-sets and countable compact sets
JO - Colloquium Mathematicae
PY - 1993
VL - 65
IS - 1
SP - 83
EP - 86
AB - In [2], D. E. Grow and M. Insall construct a countable compact set which is not the union of two H-sets. We make precise this result in two directions, proving such a set may be, but need not be, a finite union of H-sets. Descriptive set theory tools like Cantor-Bendixson ranks are used; they are developed in the book of A. S. Kechris and A. Louveau [6]. Two proofs are presented; the first one is elementary while the second one is more general and useful. Using the last one I prove in my thesis, directed by A. Louveau, the existence of a countable compact set which is not a finite union of Dirichlet sets. This result, quoted in [3], is weaker because all Dirichlet sets belong to H. Other new results about the class H and similar classes of thin sets can be found in [4], [1] and [5].
LA - eng
KW - -sets; countable compact set; Cantor-Bendixson ranks; Dirichlet sets
UR - http://eudml.org/doc/210208
ER -

References

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1. [1] H. Becker, S. Kahane and A. Louveau, Natural ${\Sigma }_{2}^{1}$ sets in harmonic analysis, Trans. Amer. Math. Soc., to appear.
2. [2] D. Grow and M. Insall, An extremal set of uniqueness?, this volume, 61-64. Zbl0838.43006
3. [3] S. Kahane, Ensembles de convergence absolue, ensembles de Dirichlet faibles et ↑-idéaux, C. R. Acad. Sci. Paris 310 (1990), 335-337.
4. [4] S. Kahane, Antistable classes of thin sets, Illinois J. Math. 37 (1) (1993). Zbl0793.42003
5. [5] S. Kahane, On the complexity of sums of Dirichlet measures, Ann. Inst. Fourier (Grenoble) 43 (1) (1993). Zbl0766.28001
6. [6] A. Kechris and A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness, London Math. Soc. Lecture Note Ser. 128, Cambridge Univ. Press, 1987.
7. [7] A. Kechris and R. Lyons, Ordinal ranking on measures annihilating thin sets, Trans. Amer. Math. Soc. 310 (1988), 747-758. Zbl0706.43007
8. [8] D. Salinger, Sur les ensembles indépendants dénombrables, C. R. Acad. Sci. Paris Sér. A-B 272 (1981), A786-788. Zbl0209.44103

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