no. 6
Two random constructions inside lacunary sets
Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1853-1867.

Nous étudions le rapport entre la croissance d’une suite d’entiers et les propriétés harmoniques et fonctionnelles de la suite de caractères associée. Nous montrons en particulier que toute suite polynomiale, ainsi que la suite des nombres premiers, contient un ensemble Λ(p) pour tout p qui n’est pas de Rosenthal.

We study the relationship between the growth rate of an integer sequence and harmonic and functional properties of the corresponding sequence of characters. In particular we show that every polynomial sequence contains a set that is Λ(p) for all p but is not a Rosenthal set. This holds also for the sequence of primes.

@article{AIF_1999__49_6_1853_0,
     author = {Neuwirth, Stefan},
     title = {Two random constructions inside lacunary sets},
     journal = {Annales de l'Institut Fourier},
     pages = {1853--1867},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
     number = {6},
     year = {1999},
     doi = {10.5802/aif.1740},
     mrnumber = {2001c:42007},
     zbl = {0955.42009},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1740/}
}
TY  - JOUR
AU  - Neuwirth, Stefan
TI  - Two random constructions inside lacunary sets
JO  - Annales de l'Institut Fourier
PY  - 1999
SP  - 1853
EP  - 1867
VL  - 49
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1740/
DO  - 10.5802/aif.1740
LA  - en
ID  - AIF_1999__49_6_1853_0
ER  - 
%0 Journal Article
%A Neuwirth, Stefan
%T Two random constructions inside lacunary sets
%J Annales de l'Institut Fourier
%D 1999
%P 1853-1867
%V 49
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1740/
%R 10.5802/aif.1740
%G en
%F AIF_1999__49_6_1853_0
Neuwirth, Stefan. Two random constructions inside lacunary sets. Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1853-1867. doi : 10.5802/aif.1740. http://www.numdam.org/articles/10.5802/aif.1740/

[1] G. Bennett, Probability inequalities for the sum of independent random variables, J. Amer. Statist. Assoc., 57 (1962), 33-45. | Zbl

[2] S.N. Bernšteĭn, On a modification of Chebyshev's inequality and on the deviation in Laplace's formula, in: Collected Works IV. Theory of probability and mathematical statistics (1911-1946), Nauka, 1964, 71-79, Russian.

[3] J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math., 61 (1988), 39-72. | MR | Zbl

[4] J. Bourgain, On Λ(p)-subsets of squares, Israel J. Math., 67 (1989), 291-311. | MR | Zbl

[5] W.J. Ellison, Les nombres premiers, Actualités Scientifiques et Industrielles 1366, Hermann, 1975. | MR | Zbl

[6] P. Erdős, Problems and results in additive number theory, in: Colloque sur la théorie des nombres (Bruxelles, 1955), Georges Thone, 1956, 127-137. | Zbl

[7] P. Erdős and A. Rényi, Additive properties of random sequences of positive integers, Acta Arith., 6 (1960), 83-110. | MR | Zbl

[8] P. Erdős and S.J. Taylor, On the set of points of convergence of a lacunary trigonometric series and the equidistribution properties of related sequences, Proc. London Math. Soc., (3) 7 (1957), 598-615. | MR | Zbl

[9] G. Godefroy, On coanalytic families of sets in harmonic analysis, Illinois J. Math., 35 (1991), 241-249. | MR | Zbl

[10] H. Halberstam and K.F. Roth, Sequences, Springer, second ed., 1983. | MR | Zbl

[11] K.E. Hare and I. Klemes, Properties of Littlewood-Paley sets, Math. Proc. Cambridge Philos. Soc., 105 (1989), 485-494. | MR | Zbl

[12] S. Karlin, Bases in Banach spaces, Duke Math. J., 15 (1948), 971-985. | MR | Zbl

[13] Y. Katznelson, Suites aléatoires d'entiers, in: L'analyse harmonique dans le domaine complexe (Montpellier, 1972), E.J. Akutowicz (ed.), Lect. Notes Math., 336, Springer, 1973, 148-152. | MR | Zbl

[14] Y. Katznelson and P. Malliavin, Un critère d'analyticité pour les algèbres de restriction, C.R. Acad. Sci. Paris, 261 (1965), 4964-4967. | MR | Zbl

[15] Y. Katznelson and P. Malliavin, Vérification statistique de la conjecture de la dichotomie sur une classe de d'algèbres de restriction, C.R. Acad. Sci. Paris, Sér. A-B, 262 (1966), A490-A492. | MR | Zbl

[16] D. Li, A remark about Λ(p)-sets and Rosenthal sets, Proc. Amer. Math. Soc., 126 (1998), 3329-3333. | MR | Zbl

[17] J.E. Littlewood and R.E.A.C. Paley, Theorems on Fourier series and power series, J. London Math. Soc., 6 (1931), 230-233. | JFM | Zbl

[18] F. Lust-Piquard, Éléments ergodiques et totalement ergodiques dans L∞(Γ), Studia Math., 69 (1981), 191-225. | MR | Zbl

[19] F. Lust-Piquard, Bohr local properties of CΛ(T), Colloq. Math., 58 (1989), 29-38. | MR | Zbl

[20] Y. Meyer, Endomorphismes des idéaux fermés de L1 (G), classes de Hardy et séries de Fourier lacunaires, Ann. sci. École Norm. Sup., (4) 1 (1968), 499-580. | Numdam | MR | Zbl

[21] Y. Meyer, Algebraic numbers and harmonic analysis, North-Holland, 1972. | MR | Zbl

[22] S. Neuwirth, Metric unconditionality and Fourier analysis, Studia Math., 131 (1998), 19-62. | MR | Zbl

[23] H.P. Rosenthal, On trigonometric series associated with weak* closed subspaces of continuous functions, J. Math. Mech., 17 (1967), 485-490. | MR | Zbl

[24] W. Rudin, Trigonometric series with gaps, J. Math. Mech., 9 (1960), 203-228. | MR | Zbl

[25] R.C. Vaughan, The Hardy-Littlewood method, Cambridge University Press, 1981. | MR | Zbl

[26] H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann., 77 (1916), 313-352. | JFM

[27] K. Zeller, Theorie der Limitierungsverfahren, Springer, 1958, Ergebnisse der Mathematik und ihrer Grenzgebiete (Neue Folge) 15. | MR | Zbl

Cité par Sources :