Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
$\sigma$-porosity; descriptive set theory; $\sigma$-ideal; trigonometric series; sets of uniqueness
$\sigma$-porosity; descriptive set theory; $\sigma$-ideal; trigonometric series; sets of uniqueness
Summary:
We show a general method of construction of non-$\sigma$-porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non-$\sigma$-porous Suslin subset of a topologically complete metric space contains a non-$\sigma$-porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non-$\sigma$-porous element. Namely, if we denote the space of all compact subsets of a compact metric space $E$ with the Vietoris topology by $\Cal K(E)$, then it is shown that each analytic subset of $\Cal K(E)$ containing all countable compact subsets of $E$ contains necessarily an element, which is a non-$\sigma$-porous subset of $E$. We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed non-$\sigma$-porous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the $\sigma$-ideal of compact $\sigma$-porous sets.
References:
[Ba] Bari N.: Trigonometric Series. Moscow, 1961. MR 0126115 | Zbl 0154.06103
[BKL] Becker H., Kahane S., Louveau A.: Some complete $\AA$ sets in harmonic analysis. Trans. Amer. Math. Soc. 339 (1993), 1 323-336. MR 1129434
[BKR] Bukovský L., Kholshchevnikova N.N., Repický M.: Thin sets of harmonic analysis and infinite combinatorics. Real Anal. Exchange 20 (1994-95), 2 454-509. MR 1348075
[De] Debs G.: Private communication.
[Do] Dolzhenko E.P.: Boundary properties of arbitrary functions. Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 3-14 (in Russian). MR 0217297
[DSR] Debs G., Saint-Raymond J.: Ensembles boréliens d'unicité au sens large. Ann. Inst. Fourier (Grenoble) 37 (1987), 3 217-239. MR 0916281
[Ka] Kaufman R.: Fourier transforms and descriptive set theory. Mathematika 31 (1984), 2 336-339. MR 0804207 | Zbl 0604.42009
[Ke] Kechris A.S.: Classical Descriptive Set Theory. Springer-Verlag, New York, 1995. MR 1321597 | Zbl 0819.04002
[KL] Kechris A.S., Louveau A.: Descriptive Set Theory and the Structure of Sets of Uniqueness. London Math. Soc. Lecture Notes Series 128, Cambridge University Press, Cambridge, 1989. MR 0953784 | Zbl 0677.42009
[KLW] Kechris A.S., Louveau A., Woodin W.H.: The structure of $\sigma$-ideals of compact sets. Trans. Amer. Math. Soc. 301 (1987), 1 263-288. MR 0879573 | Zbl 0633.03043
[La] Laczkovich M.: Analytic subgroups of the reals. Proc. Amer. Math. Soc. 126 (1998), 6 1783-1790. MR 1443837 | Zbl 0896.04002
[Lo] Loomis L.: The spectral characterization of a class of almost periodic functions. Ann. of Math. 72 (1960), 2 362-368. MR 0120502 | Zbl 0094.05801
[LP] Lindahl L.-A., Poulsen F.: Thin Sets in Harmonic Analysis. Marcel Dekker, New York, 1971. MR 0393993 | Zbl 0226.43006
[PS] Piatetski-Shapiro I.I.: On the problem of uniqueness expansion of a function in a trigonometric series. Moscov. Gos. Univ. Uchen. Zap., vol. 155, Mat. 5 (1952), 54-72. MR 0080201
[R] Rogers C.A. et al.: Analytic Sets. Academic Press, London, 1980. MR 0608794 | Zbl 0589.54047
[Re] Reclaw I.: A note on the $\sigma$-ideal of $\sigma$-porous sets. Real Anal. Exchange 12 (1986-87), 2 455-457. MR 0888722 | Zbl 0656.26001
[So] Solecki S.: Covering analytic sets by families of closed sets. J. Symbolic Logic 59 (1994), 3 1022-1031. MR 1295987 | Zbl 0808.03031
[Šl] Šleich P.: Sets of type $H^{(s)}$ are $\sigma$-bilaterally porous. preprint (unpublished).
[Za$_1$] Zajíček L.: Sets of $\sigma$-porosity and $\sigma$-porosity $(q)$. Časopis Pěst. Mat. 101 (1976), 4 350-359. MR 0457731
[Za$_2$] Zajíček L.: Porosity and $\sigma$-porosity. Real Anal. Exchange 13 (1987-88), 2 314-350. MR 0943561
[Za$_3$] Zajíček L.: Small non-sigma-porous sets in topologically complete metric spaces. Colloq. Math. 77 (1998), 2 293-304. MR 1628994
[Za$_4$] Zajíček L.: Smallness of sets of nondifferentiability of convex functions in non-separable Banach spaces. Czechoslovak Math. J. 41 (116) (1991), 288-296. MR 1105445
[Za$_5$] Zajíček L.: An unpublished result of P. Sleich: sets of type $H^{(s)}$ are $\sigma$-bilaterally porous. Real Anal. Exchange 27 (2002), 1 363-372. MR 1887868
[Ze$_1$] Zelený M.: Calibrated thin $\boldsymbol\Pi_{\bold 1}^{\bold 1}$ $\sigma$-ideals are $\boldsymbol G_{\delta}$. Proc. Amer. Math. Soc. 125 (1997), 10 3027-3032. MR 1415378
[Ze$_2$] Zelený M.: On singular boundary points of complex functions. Mathematika 45 (1998), 1 119-133. MR 1644354
Search
Partner of
