Two random constructions inside lacunary sets

Stefan Neuwirth

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 6, page 1853-1867
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Neuwirth, Stefan. "Two random constructions inside lacunary sets." Annales de l'institut Fourier 49.6 (1999): 1853-1867. <http://eudml.org/doc/75404>.

@article{Neuwirth1999,
abstract = {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 $\Lambda (p)$ for all $p$ but is not a Rosenthal set. This holds also for the sequence of primes.},
author = {Neuwirth, Stefan},
journal = {Annales de l'institut Fourier},
keywords = {-set; Rosenthal set; equidistributed set of integers; uniformly distributed set of integers},
language = {eng},
number = {6},
pages = {1853-1867},
publisher = {Association des Annales de l'Institut Fourier},
title = {Two random constructions inside lacunary sets},
url = {http://eudml.org/doc/75404},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Neuwirth, Stefan
TI - Two random constructions inside lacunary sets
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 6
SP - 1853
EP - 1867
AB - 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 $\Lambda (p)$ for all $p$ but is not a Rosenthal set. This holds also for the sequence of primes.
LA - eng
KW - -set; Rosenthal set; equidistributed set of integers; uniformly distributed set of integers
UR - http://eudml.org/doc/75404
ER -

References

top
  1. [1] G. BENNETT, Probability inequalities for the sum of independent random variables, J. Amer. Statist. Assoc., 57 (1962), 33-45. Zbl0104.11905
  2. [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. [3] J. BOURGAIN, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math., 61 (1988), 39-72. Zbl0642.28010MR89f:28037a
  4. [4] J. BOURGAIN, On Λ(p)-subsets of squares, Israel J. Math., 67 (1989), 291-311. Zbl0692.43005MR91d:43018
  5. [5] W.J. ELLISON, Les nombres premiers, Actualités Scientifiques et Industrielles 1366, Hermann, 1975. Zbl0313.10001MR54 #5138
  6. [6] P. ERDŐS, Problems and results in additive number theory, in: Colloque sur la théorie des nombres (Bruxelles, 1955), Georges Thone, 1956, 127-137. Zbl0073.03102
  7. [7] P. ERDŐS and A. RÉNYI, Additive properties of random sequences of positive integers, Acta Arith., 6 (1960), 83-110. Zbl0091.04401MR22 #10970
  8. [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. Zbl0111.26801MR19,1050b
  9. [9] G. GODEFROY, On coanalytic families of sets in harmonic analysis, Illinois J. Math., 35 (1991), 241-249. Zbl0703.43005MR92b:43011
  10. [10] H. HALBERSTAM and K.F. ROTH, Sequences, Springer, second ed., 1983. Zbl0498.10001MR83m:10094
  11. [11] K.E. HARE and I. KLEMES, Properties of Littlewood-Paley sets, Math. Proc. Cambridge Philos. Soc., 105 (1989), 485-494. Zbl0691.43006MR90f:42018
  12. [12] S. KARLIN, Bases in Banach spaces, Duke Math. J., 15 (1948), 971-985. Zbl0032.03102MR10,548c
  13. [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. Zbl0274.42009MR53 #1176
  14. [14] Y. KATZNELSON and P. MALLIAVIN, Un critère d'analyticité pour les algèbres de restriction, C.R. Acad. Sci. Paris, 261 (1965), 4964-4967. Zbl0143.36001MR34 #3226a
  15. [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. Zbl0143.36002MR34 #3226b
  16. [16] D. LI, A remark about Λ(p)-sets and Rosenthal sets, Proc. Amer. Math. Soc., 126 (1998), 3329-3333. Zbl0907.43007MR99a:43005
  17. [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. Zbl0002.18803JFM57.0318.01
  18. [18] F. LUST-PIQUARD, Éléments ergodiques et totalement ergodiques dans L∞(Γ), Studia Math., 69 (1981), 191-225. Zbl0476.43001MR84h:43003
  19. [19] F. LUST-PIQUARD, Bohr local properties of CΛ(T), Colloq. Math., 58 (1989), 29-38. Zbl0694.43005MR91c:43009
  20. [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. Zbl0169.18001MR39 #1910
  21. [21] Y. MEYER, Algebraic numbers and harmonic analysis, North-Holland, 1972. Zbl0267.43001MR58 #5579
  22. [22] S. NEUWIRTH, Metric unconditionality and Fourier analysis, Studia Math., 131 (1998), 19-62. Zbl0930.42005MR99f:42017
  23. [23] H.P. ROSENTHAL, On trigonometric series associated with weak* closed subspaces of continuous functions, J. Math. Mech., 17 (1967), 485-490. Zbl0194.16703MR35 #7064
  24. [24] W. RUDIN, Trigonometric series with gaps, J. Math. Mech., 9 (1960), 203-228. Zbl0091.05802MR22 #6972
  25. [25] R.C. VAUGHAN, The Hardy-Littlewood method, Cambridge University Press, 1981. Zbl0455.10034MR84b:10002
  26. [26] H. WEYL, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann., 77 (1916), 313-352. Zbl46.0278.06JFM46.0278.06
  27. [27] K. ZELLER, Theorie der Limitierungsverfahren, Springer, 1958, Ergebnisse der Mathematik und ihrer Grenzgebiete (Neue Folge) 15. Zbl0085.04603MR22 #9759

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.