Finite union of H-sets and countable compact sets
Kahane, Sylvain
Colloquium Mathematicae, Tome 66 (1993), p. 83 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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].

Publié le : 1993-01-01
EUDML-ID : urn:eudml:doc:210208 
@article{bwmeta1.element.bwnjournal-article-cmv65i1p83bwm,
     author = {Sylvain Kahane},
     title = {Finite union of H-sets and countable compact sets},
     journal = {Colloquium Mathematicae},
     volume = {66},
     year = {1993},
     pages = {83},
     zbl = {0865.43007},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv65i1p83bwm}
}

Kahane, Sylvain. Finite union of H-sets and countable compact sets. Colloquium Mathematicae, Tome 66 (1993) p. 83. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv65i1p83bwm/

[000] [1] H. Becker, S. Kahane and A. Louveau, Natural Σ21 sets in harmonic analysis, Trans. Amer. Math. Soc., to appear.

[001] [2] D. Grow and M. Insall, An extremal set of uniqueness?, this volume, 61-64. | Zbl 0838.43006

[002] [3] S. Kahane, Ensembles de convergence absolue, ensembles de Dirichlet faibles et ↑-idéaux, C. R. Acad. Sci. Paris 310 (1990), 335-337.

[003] [4] S. Kahane, Antistable classes of thin sets, Illinois J. Math. 37 (1) (1993). | Zbl 0793.42003

[004] [5] S. Kahane, On the complexity of sums of Dirichlet measures, Ann. Inst. Fourier (Grenoble) 43 (1) (1993). | Zbl 0766.28001

[005] [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.

[006] [7] A. Kechris and R. Lyons, Ordinal ranking on measures annihilating thin sets, Trans. Amer. Math. Soc. 310 (1988), 747-758. | Zbl 0706.43007

[007] [8] D. Salinger, Sur les ensembles indépendants dénombrables, C. R. Acad. Sci. Paris Sér. A-B 272 (1981), A786-788. | Zbl 0209.44103