Skip to main content
SpringerLink
Log in

Some new thin sets of integers in harmonic analysis

  • Published:
Journal d’Analyse Mathématique Aims and scope

Abstract

We randomly construct various subsets A of the integers which have both smallness and largeness properties. They are small since they are very close, in various senses, to Sidon sets: the continuous functions with spectrum in Λ have uniformly convergent series, and their Fourier coefficients are in ℓp for all p > 1; moreover, all the Lebesgue spaces L qΛ are equal forq < +∞. On the other hand, they are large in the sense that they are dense in the Bohr group and that the space of the bounded functions with spectrum in Λ is nonseparable. So these sets are very different from the thin sets of integers previously known.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. I. Arkipov and K. I. Oskolkov,On a special trigonometric series and its applications, Math. Sbornik134 (1987), 145–155.

    Google Scholar 

  2. R. Blei,Sidon partitions and p-Sidon sets, Pacific J. Math.65 (1976), 307–313.

    MATH  MathSciNet  Google Scholar 

  3. R. Blei,Fractional cartesian products of sets, Ann. Inst. Fourier (Grenoble)29 (1979), 79–105.

    MATH  MathSciNet  Google Scholar 

  4. R. Blei,Combinatorial dimension and certain norms in harmonic analysis, Amer. J. Math.106 (1984), 847–887.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Blei and T. W. Körner,Combinatorial dimension and random sets, Israel J. Math.47 (1984), 65–74.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. R. Blum, B. Eisenberg and L.-S. Hahn,Ergodic theory and the measure of sets in the Bohr group, Acta Sci. Math. (Szeged)34 (1973), 17–24.

    MATH  MathSciNet  Google Scholar 

  7. J. Bourgain,Propriétés de décomposition pour les ensembles de Sidon, Bull. Soc. Math. France111 (1983), 421–428.

    MATH  MathSciNet  Google Scholar 

  8. J. Bourgain,Sidon sets and Riesz products, Ann. Inst. Fourier (Grenoble)35 (1985), 137–148.

    MATH  MathSciNet  Google Scholar 

  9. J. Bourgain, Subspaces of {LskN/∞}, arithmetical diameter and Sidon sets, inProbability in Banach Spaces V, Proc. Medford 1984, Lecture Notes in Math. 1153, Springer-Verlag, Berlin, 1985, pp. 96–127.

    Chapter  Google Scholar 

  10. J. Bourgain,On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math.61 (1988), 39–72.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Bourgain,On the behavior of the constant in the Littlewood-Paley inequality, inGeometric Aspects of Functional Analysis (1987–88), Lecture Notes in Math. 1376, Springer-Verlag, Berlin, 1989, pp. 202–208.

    Google Scholar 

  12. J. Bourgain,Bounded orthogonal sets and the Λ(p)-set problem, Acta Math.162 (1989), 227–246.

    Article  MathSciNet  Google Scholar 

  13. J. Bourgain and V. Milman,Dichotomie du cotype pour les espaces invariants, C. R. Acad. Sci. Paris300 (1985), 263–266.

    MATH  MathSciNet  Google Scholar 

  14. M. Bożejko and T. Pytlik,Some types of lacunary Fourier series, Colloq. Math.25 (1972), 117–124.

    MathSciNet  Google Scholar 

  15. M. Déchamps-Gondim,Sur les compacts associés aux ensembles lacunaires, les ensembles de Sidon et quelques problèmes ouverts, Publications Mathématiques d’Orsay 84-01 (1984), exposé 7.

  16. J. Diestel,Sequences and Series in Banach Spaces, Springer-Verlag, Berlin, 1984.

    Google Scholar 

  17. S. Drury,Sur les ensembles de Sidon, C. R. Acad. Sci. Paris271 (1970), 162–163.

    MATH  MathSciNet  Google Scholar 

  18. R. E. Edwards and K. A. Ross,p-Sidon sets, J. Funct. Anal.15 (1974), 404–427.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Figà-Talamanca,An example in the theory of lacunary Fourier series, Boll. Un. Mat. Ital.3 (1970), 375–378.

    MATH  MathSciNet  Google Scholar 

  20. J. Fournier,Two UC-sets whose union is not a UC-set, Proc. Amer. Math. Soc.84 (1982), 69–72.

    Article  MATH  MathSciNet  Google Scholar 

  21. J. Fournier and L. Pigno,Analytic and arithmetic properties of thin sets, Pacific J. Math.105 (1983), 115–141.

    MATH  MathSciNet  Google Scholar 

  22. C. Frappier, Q. I. Rahman and St. Ruscheweyh,New inequalities for polynomials, Trans. Amer. Math. Soc.288 (1985), 69–99.

    Article  MATH  MathSciNet  Google Scholar 

  23. G. W. Johnson,Theorems on lacunary sets, especially p-Sidon sets, Studia Math.58 (1976), 209–221.

    MATH  MathSciNet  Google Scholar 

  24. G. W. Johnson and G. S. Woodward,On p-Sidon sets, Indiana Univ. Math. J.24 (1974), 161–167.

    Article  MATH  MathSciNet  Google Scholar 

  25. J.-P. Kahane,Some Random Series of Functions, second edn., Cambridge Univ. Press, Cambridge, 1985.

    MATH  Google Scholar 

  26. B. Kashin and L. Tzafriri,On random sets of uniform convergence, Math. Notes54 (1993), 677–687.

    Article  MATH  MathSciNet  Google Scholar 

  27. Y. Katznelson,Suites aléatoires d’entiers, inL’analyse harmonique dans le domaine complexe (Actes Table Ronde Internat. Centre Nat. Recherche Sci., Montpellier 1972), Lecture Notes in Math.336, Springer-Verlag, Berlin, 1973, pp. 148–152.

    Google Scholar 

  28. Y. Katznelson and P. Malliavin,Vérification statistique de la conjecture de la dichotomie sur une classe d’algèbres de restriction, C. R. Acad. Sci. Paris262 (1966), 490–492.

    MATH  MathSciNet  Google Scholar 

  29. S. Konyagin,On divergence of trigonometric Fourier series everywhere, C. R. Acad. Sci. Paris329 (1999), 693–697.

    MATH  MathSciNet  Google Scholar 

  30. T. W. Körner,Fourier Analysis, Cambridge University Press, Cambridge, 1988.

    MATH  Google Scholar 

  31. S. Kwapien and A. Pełczyński,Absolutely summing operators and translation invariant spaces of functions on compact abelian groups, Math. Nachr.94 (1980), 303–340.

    Article  MATH  MathSciNet  Google Scholar 

  32. M. Ledoux and M. Talagrand,Probability in Banach Spaces, Springer-Verlag, Berlin, 1991.

    MATH  Google Scholar 

  33. P. Lefèvre,Sur les ensembles de convergence uniforme, Publ. Math. Orsay94-24 (1994), 1–70.

    Google Scholar 

  34. P. Lefèvre,Measures and lacunarity sets, Studia Math.133 (1999), 145–161.

    MATH  MathSciNet  Google Scholar 

  35. P. Lefèvre, D. Li, H. Queffélec, and L. Rodríguez-Piazza,Lacunary sets and function spaces with finite cotype, J. Funct. Anal., to appear.

  36. P. Lefèvre and L. Rodríguez-Piazza,p-Rider sets are q-Sidon sets, submitted.

  37. D. Li,A remark about Λ(p)-sets and Rosenthal sets, Proc. Amer. Math. Soc.126 (1998) 3329–3333.

    Article  MATH  MathSciNet  Google Scholar 

  38. J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces I and II, Springer-Verlag, Berlin, 1997.

    Google Scholar 

  39. J. M. Lopez and K. A. Ross,Sidon Sets, Marcel Dekker, New York, 1975.

    MATH  Google Scholar 

  40. F. Lust,Produits tensoriels injectifs d’espaces de Sidon, Colloq. Math.32 (1975), 285–289.

    MATH  MathSciNet  Google Scholar 

  41. F. Lust-Piquard,Propriétés géométriques des sous-espaces invariants par translation de L1(G)et C(G), Sém. Géom. Espaces Banach, Ecole Polytechnique, Paris (1977–78), Exposé n°26.

  42. F. Lust-Piquard,Bohr local properties of CgL(T), Colloq. Math.58 (1989), 29–38.

    MATH  MathSciNet  Google Scholar 

  43. S. Neuwirth,Random constructions inside lacunary sets, Ann. Inst. Fourier (Grenoble)49 (1999), 1853–1867.

    MATH  MathSciNet  Google Scholar 

  44. K. I. Oskolkov,On spectra of uniform convergence, Soviet Math. Dokl.33 (1986), 616–620.

    MATH  Google Scholar 

  45. L. Pedemonte,Sets of uniform convergence, Colloq. Math.33 (1975), 123–132.

    MATH  MathSciNet  Google Scholar 

  46. G. Pisier,Ensembles de Sidon et espaces de cotype 2, Séminaire sur la géométrie des espaces de Banach 1977–1978, Ecole Polytechnique, Paris (1978), exposé 14.

    Google Scholar 

  47. G. Pisier,Sur l’espace de Banach des séries de Fourier aléatoires presque sûrement continues. Séminaire sur la géométrie des espaces de Banach 1977–1978, Ecole Polytechnique, Paris (1978), exposés 17–18.

    Google Scholar 

  48. G. Pisier,De nouvelles caractérisatlons des ensembles de Sidon, Math. Anal. Appl., Part B, Advances in Math. Suppl. Studies, Vol.7B (1981), 685–726.

    MathSciNet  Google Scholar 

  49. D. Rider,Gap series on groups and spheres, Canad. J. Math.18 (1966), 389–398.

    MATH  MathSciNet  Google Scholar 

  50. D. Rider,Randomly continuous functions and Sidon sets, Duke Math. J.42 (1975), 759–764.

    Article  MATH  MathSciNet  Google Scholar 

  51. L. Rodríguez-Piazza,Caractérisatlon des ensembles p-Sidon p.s., C. R. Acad. Sci. Paris305 (1987), 237–240.

    MATH  Google Scholar 

  52. L. Rodríguez-Piazza,Rango y propiedades de medidas vectoriales. Conjuntos p-Sldon p.s., Thesis, Universidad de Sevilla, 1991.

  53. H. P. Rosenthal,On trigonometric series associated with weak* closed subspaces of continuous functions, J. Math. Mech.17 (1967), 485–490.

    MATH  MathSciNet  Google Scholar 

  54. W. Rudin,Trigonometric series with gaps, J. Math. Mech.9 (1960), 203–227.

    MATH  MathSciNet  Google Scholar 

  55. I. Singer,Bases in Banach Spaces I, Springer-Verlag, Berlin, 1970.

    MATH  Google Scholar 

  56. P. M. Soardi and G. Travaglini,On sets of completely uniform convergence, Colloq. Math.45 (1981), 317–320.

    MATH  MathSciNet  Google Scholar 

  57. G. Travaglini,Some properties of UC-sets, Boll. Un. Mat. Ital.15 (1978), 272–284.

    MATH  MathSciNet  Google Scholar 

  58. N. T. Varopoulos,Une remarque sur les ensembles de Helson, Duke Math. J.43 (1976), 387–390.

    Article  MATH  MathSciNet  Google Scholar 

  59. P. Wojtaszczyk,Banach Spaces for Analysts, Cambridge University Press, Cambridge, 1991.

    MATH  Google Scholar 

  60. G. S. Woodward,p-Sidon sets and a uniform property, Indiana Univ. Math. J.25 (1976), 995–1003.

    Article  MATH  MathSciNet  Google Scholar 

  61. Z. Zalcwasser,Polynômes associés aux fonctions modulaires ϑ, Studia Math.7 (1938), 16–35.

    MATH  Google Scholar 

  62. A. Zygmund,Trigonometric Series, second edn. Vols. I & II, Cambridge Univ. Press, Cambridge, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Pôle de Lens Faculté des Sciences Jean Perrin, Université d’Artois, rue Jean Souvraz SP 18, F-62307, Lens Cedex, France

    D. Li

  2. UFR de Mathématiques, Université des Sciences et Technologies de Lille, F-59655, Villeneuve d’Ascq Cedex, France

    H. Queffélec

  3. Facultad de Matematicas Departamento de Análisis Matemático, Universidad de Sevilla, Apartado de Correos 1160, 41080, Sevilla, Spain

    L. Rodríguez-Piazza

Authors
  1. D. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Queffélec
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Rodríguez-Piazza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, D., Queffélec, H. & Rodríguez-Piazza, L. Some new thin sets of integers in harmonic analysis. J. Anal. Math. 86, 105–138 (2002). https://doi.org/10.1007/BF02786646

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02786646

Keywords

Search

Navigation