I0 Sets and Their Characterizations

  • Colin C. Graham
  • Kathryn E. Hare
Part of the CMS Books in Mathematics book series (CMSBM)


I 0 sets characterized analytically, in terms of function algebras, and topologically. I 0 sets do not cluster at continuous characters.


  1. [1]
    F. Albiac and N. J. Kalton. Topics in Banach Space Theory, volume 233 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 2006.Google Scholar
  2. [2]
    M. E. Andersson. The Kaufman-Rickert inequality governs Rademacher sums. Analysis (Munich), 23:65–79, 2003.Google Scholar
  3. [3]
    J. Arias de Reyna. Pointwise Convergence of Fourier Series, volume 1785 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 2002.Google Scholar
  4. [4]
    N. Asmar and S. Montgomery-Smith. On the distribution of Sidon series. Arkiv för Math., 31(1):13–26, 1993.Google Scholar
  5. [5]
    G. Bachelis and S. Ebenstein. On Λ(p) sets. Pacific J. Math., 54:35–38, 1974.Google Scholar
  6. [6]
    S. Banach. Über einige Eigenschaften der lakunäre trigonometrischen Reihen, II. Studia Math., 2:207–220, 1930.Google Scholar
  7. [7]
    N. K. Bary. A Treatise on Trigonometric Series, volume I. MacMillan, New York, N. Y., 1964.Google Scholar
  8. [8]
    F. Baur. Lacunarity on nonabelian groups and summing operators. J. Aust. Math. Soc., 71(1):71–79, 2001.Google Scholar
  9. [9]
    G. Benke. Arithmetic structure and lacunary Fourier series. Proc. Amer. Math. Soc., 34:128–132, 1972.Google Scholar
  10. [10]
    G. Benke. On the hypergroup structure of central Λ(p) sets. Pacific J. Math., 50:19–27, 1974.Google Scholar
  11. [11]
    D. Berend. Parallelepipeds in sets of integers. J. Combin. Theory, 45 (2):163–170, 1987.Google Scholar
  12. [12]
    K. G. Binmore. Analytic functions with Hadamard gaps. Bull. London Math. Soc., 1:211–217, 1969.Google Scholar
  13. [13]
    R. C. Blei. On trigonometric, series associated with separable, translation invariant subspaces of L . Trans. Amer. Math. Soc., 173:491–499, 1972.Google Scholar
  14. [14]
    R. C. Blei. Fractional Cartesian products of sets. Ann. Inst. Fourier (Grenoble), 29(2):79–105, 1979.Google Scholar
  15. [15]
    S. Bochner. Monotone Funktionen, Stieltjessche Integrale, und harmonische Analyse. Math. Ann., 108:378–410, 1933.Google Scholar
  16. [16]
    A. Bonami. Étude des coefficients de Fourier des fonctions de L p(G). Ann. Inst. Fourier (Grenoble), 20(2):335–402, 1970.Google Scholar
  17. [17]
    J. Bourgain. Propriétés de décomposition pour les ensembles de Sidon. Bull. Soc. Math. France, 111(4):421–428, 1983.Google Scholar
  18. [18]
    J. Bourgain. Subspaces of N , arithmetical diameter and Sidon sets. In Probability in Banach spaces, V (Medford, Mass., 1984), volume 1153 of Lecture Notes in Math., pages 96–127, Berlin, Heidelberg, New York, 1985. Springer.Google Scholar
  19. [19]
    J. Bourgain. Sidon sets and Riesz products. Ann. Inst. Fourier (Grenoble), 35(1):137–148, 1985.Google Scholar
  20. [20]
    J. Bourgain. A remark on entropy of abelian groups and the invariant uniform approximation property. Studia Math., 86:79–84, 1987.Google Scholar
  21. [21]
    J. Bourgan and V. Milḿan. Dichotomie du cotype pours les espace invariants. C. R. Acad. Sci. Paris, pages 435–438, 1985.Google Scholar
  22. [22]
    L Carleson. On convergence and growth of partial sums of Fourier series. Acta Math., 116:135–157, 1966.Google Scholar
  23. [23]
    D. Cartwright and J. McMullen. A structural criterion for Sidon sets. Pacific J. Math., 96:301–317, 1981.Google Scholar
  24. [24]
    Mei-Chu Chang. On problems of Erdös and Rudin. J. Functional Anal., 207(2):444–460, 2004.Google Scholar
  25. [25]
    Y. S. Chow and H. Teicher. Probability Theory. Independence, Interchangeability, Martingales. Springer Texts in Statistics. Springer-Verlag, Berlin, Heidelberg, New York, 3rd edition, 1997.Google Scholar
  26. [26]
    E. Crevier. Private communication. 2012.Google Scholar
  27. [27]
    M. Déchamps(-Gondim). Ensembles de Sidon topologiques. Ann. Inst. Fourier (Grenoble), 22(3):51–79, 1972.Google Scholar
  28. [28]
    M. Déchamps(-Gondim). Densité harmonique et espaces de Banach ne contenant pas de sous-espace fermé isomorphe à c 0. C. R. Acad. Sci. Paris, 282(17):A963–A965, 1976.Google Scholar
  29. [29]
    M. Déchamps(-Gondim). Densité harmonique et espaces de Banach invariants par translation ne contenant pas c 0. Colloquium Math., 51:67–84, 1987.Google Scholar
  30. [30]
    M. Déchamps(-Gondim) and O. Selles. Compacts associés aus sommes de suites lacunaires. Publ. Math. Orsay, 1:27–40, 1996.Google Scholar
  31. [31]
    R. Doss. Elementary proof of a theorem of Helson. Proc. Amer. Math. Soc., 27(2):418–420, 1971.Google Scholar
  32. [32]
    S. W. Drury. Sur les ensembles de Sidon. C. R. Acad. Sci. Paris, 271A: 162–163, 1970.Google Scholar
  33. [33]
    S. W. Drury. The Fatou-Zygmund property for Sidon sets. Bull. Amer. Math. Soc., 80:535–538, 1974.Google Scholar
  34. [34]
    W. F. Eberlein. Characterizations of Fourier-Stieltjes transforms. Duke Math. J., 22:465–468, 1955.Google Scholar
  35. [35]
    R. E. Edwards and K. A. Ross. p-Sidon sets. J. Functional Anal., 15:404–427, 1974.Google Scholar
  36. [36]
    R. E. Edwards, E. Hewitt, and K. A. Ross. Lacunarity for compact groups, III. Studia Math., 44:429–476, 1972.Google Scholar
  37. [37]
    W. Feller. An Introduction to Probability Theory and its Applications. John Wiley and Sons, New York, London, 2nd edition, 1957.Google Scholar
  38. [38]
    A. Figá-Talamanca and D. G. Rider. A theorem of Littlewood and lacunary series for compact groups. Pacific J. Math., 16:505–514, 1966.Google Scholar
  39. [39]
    John J. F. Fournier and Louis Pigno. Analytic and arithmetic properties of thin sets. Pacific J. Math., 105(1):115–141, 1983.Google Scholar
  40. [40]
    W. H. J. Fuchs. On the zeros of a power series with Hadamard gaps. Nagoya Math. J., 29:167–174, 1967.Google Scholar
  41. [41]
    J. Galindo and S. Hernández. The concept of boundedness and the Bohr compactification of a MAP abelian group. Fund. Math., 159(3): 195–218, 1999.Google Scholar
  42. [42]
    J. Galindo and S. Hernández. Interpolation sets and the Bohr topology of locally compact groups. Adv. in Math., 188:51–68, 2004.Google Scholar
  43. [43]
    I. M. Gel’fand. Normierte Ringe. Mat. Sb., N.S., 9:3–24, 1941.Google Scholar
  44. [44]
    J. Gerver. The differentiability of the Riemann function at certain rational multiples of π. Amer. J. Math., 92:33–55, 1970.Google Scholar
  45. [45]
    J. Gerver. More on the differentiability of the Riemann function. Amer. J. Math., 93:33–41, 1971.Google Scholar
  46. [46]
    B. N. Givens and K. Kunen. Chromatic numbers and Bohr topologies. Topology Appl., 131(2):189–202, 2003.Google Scholar
  47. [47]
    D. Gnuschke and Ch. Pommerenke. On the radial limits of functions with Hadamard gaps. Michigan Math. J., 32(1):21–31, 1985.Google Scholar
  48. [48]
    E. Goursat. A Course in Mathematical Analysis, volume I. Ginn & Co., Boston, Chicago, London, New York, 1904. E. R. Hedrick, trans.Google Scholar
  49. [49]
    W. T. Gowers. A new proof of Szemeredi’s theorem. Geom. Functional Anal., 11:465–588, 2001.Google Scholar
  50. [50]
    C. C. Graham. Sur un théorème de Katznelson et McGehee. C. R. Acad. Sci. Paris, 276:A37–A40, 1973.Google Scholar
  51. [51]
    C. C. Graham and K. E. Hare. ε-Kronecker and I 0 sets in abelian groups, III: interpolation by measures on small sets. Studia Math., 171(1):15–32, 2005.Google Scholar
  52. [52]
    C. C. Graham and K. E. Hare. ε-Kronecker and I 0 sets in abelian groups, I: arithmetic properties of ε-Kronecker sets. Math. Proc.Cambridge Philos. Soc., 140(3):475–489, 2006.Google Scholar
  53. [53]
    C. C. Graham and K. E. Hare. ε-Konecker and I 0 sets in abelian groups, IV: interpolation by non-negative measures. Studia Math., 177(1):9–24, 2006.Google Scholar
  54. [54]
    C. C. Graham and K. E. Hare. I 0 sets for compact, connected groups: interpolation with measures that are nonnegative or of small support. J. Austral. Math. Soc., 84(2): 199–225, 2008.Google Scholar
  55. [55]
    C. C. Graham and K. E. Hare. Characterizing Sidon sets by interpolation properties of subsets. Colloquium Math., 112(2):175–199, 2008.Google Scholar
  56. [56]
    C. C. Graham and O. C. McGehee. Essays in Commutative Harmonic Analysis. Number 228 in Grundleheren der Mat. Wissen. Springer-Verlag, Berlin, Heidelberg, New York, 1979.Google Scholar
  57. [57]
    C. C. Graham and K. E. Hare. Characterizations of some classes of I 0 sets. Rocky Mountain J. Math., 40(2):513–525, 2010.Google Scholar
  58. [58]
    C. C. Graham and K. E. Hare. Sets of zero discrete harmonic density. Math. Proc. Cambridge Philos. Soc., 148(2):253–266, 2010.Google Scholar
  59. [59]
    C. C. Graham and K. E. Hare. Existence of large ε-Kronecker and FZI0(U) sets in discrete abelian groups. Colloquium Math., 2012.Google Scholar
  60. [60]
    C. C. Graham and A. T.-M. Lau. Relative weak compactness of orbits in Banach spaces associated with locally compact groups. Trans. Amer. Math. Soc., 359:1129–1160, 2007.Google Scholar
  61. [61]
    C. C. Graham, K. E. Hare, and T. W. Körner. ε-Kronecker and I 0 sets in abelian groups, II: sparseness of products of ε-Kronecker sets. Math. Proc. Cambridge Philos. Soc., 140(3):491–508, 2006.Google Scholar
  62. [62]
    C. C. Graham, K. E. Hare, and (L.) T. Ramsey. Union problems for I 0 sets. Acta Sci. Math. (Szeged), 75(1–2):175–195, 2009.Google Scholar
  63. [63]
    C. C. Graham, K. E. Hare, and (L.) T. Ramsey. Union problems for I 0 sets-corrigendum. Acta Sci. Math. (Szeged), 76(3–4):487–8, 2009.Google Scholar
  64. [64]
    D. Grow. A class of I 0-sets. Colloquium Math., 53(1):111–124, 1987.Google Scholar
  65. [65]
    D. Grow. Sidon sets and I 0-sets. Colloquium Math., 53(2):269–270, 1987.Google Scholar
  66. [66]
    D. Grow. A further note on a class of I 0-sets. Colloquium Math., 53(1):125–128, 1987.Google Scholar
  67. [67]
    D. Grow and K. E. Hare. The independence of characters on non-abelian groups. Proc. Amer. Math. Soc., 132(12):3641–3651, 2004.Google Scholar
  68. [68]
    D. Grow and K. E. Hare. Central interpolation sets for compact groups and hypergroups. Glasgow Math. J., 51(3):593–603, 2009.Google Scholar
  69. [69]
    A. Gut. An Intermediate Course in Probability. Springer Texts in Statistics. Springer, Berlin, Heidelberg, New York, 2nd edition, 2009.Google Scholar
  70. [70]
    J. Hadamard. Essai sur l’étude des fonctions données par leur développement de Taylor. J. Math. Pures Appl., 8(4):101–186, 1892.Google Scholar
  71. [71]
    A. Harcharras. Fourier analysis, Schur multipliers on S p and non-commutative Λ(p)-sets. Studia Math., 137(3):203–260, 1999.Google Scholar
  72. [72]
    G. H. Hardy. Weierstrass’s non-differentiable function. Trans. Amer. Math. Soc., 17:301–325, 1916.Google Scholar
  73. [73]
    K. E. Hare. Arithmetic properties of thin sets. Pacific J. Math., 131: 143–155, 1988.Google Scholar
  74. [74]
    K. E. Hare. An elementary proof of a result on Λ(p) sets. Proc. Amer. Math. Soc., 104:829–832, 1988.Google Scholar
  75. [75]
    K. E. Hare and (L.) T. Ramsey. I 0 sets in non-abelian groups. Math. Proc. Cambridge Philos. Soc., 135: 81–98, 2003.Google Scholar
  76. [76]
    K. E. Hare and (L.) T. Ramsey. Kronecker constants for finite subsets of integers. J. Fourier Anal. Appl., 18(2):326–366, 2012.Google Scholar
  77. [77]
    K. E. Hare and N. Tomczak-Jaegermann. Some Banach space properties of translation-invariant subspaces of L p. In Analysis at Urbana, I, 1986-1987, volume 137 of London Math. Soc. Lecture Notes, pages 185–195. Cambridge Univ. Press, Cambridge, U. K., 1989.Google Scholar
  78. [78]
    K. E. Hare and D. C. Wilson. A structural criterion for the existence of infinite central Λ(p) sets. Trans. Amer. Math. Soc., 337(2):907–925, 1993.Google Scholar
  79. [79]
    S. Hartman. On interpolation by almost periodic functions. Colloquium Math, 8:99–101, 1961.Google Scholar
  80. [80]
    S. Hartman. Interpolation par les mesures diffuses. Colloquium Math., 26:339–343, 1972.Google Scholar
  81. [81]
    S. Hartman and C. Ryll-Nardzewski. Almost periodic extensions of functions. Colloquium Math., 12:23–39, 1964.Google Scholar
  82. [82]
    S. Hartman and C. Ryll-Nardzewski. Almost periodic extensions of functions, II. Colloquium Math., 15:79–86, 1966.Google Scholar
  83. [83]
    S. Hartman and C. Ryll-Nardzewski. Almost periodic extensions of functions, III. Colloquium Math., 16:223–224, 1967.Google Scholar
  84. [84]
    H. Helson. Fourier transforms on perfect sets. Studia. Math., 14:209–213, 1954.Google Scholar
  85. [85]
    H. Helson and J.-P. Kahane. A Fourier method in diophantine problems. J. d’Analyse Math., 15:245–262, 1965.Google Scholar
  86. [86]
    C. Herz. Drury’s lemma and Helson sets. Studia Math., 42:205–219, 1972.Google Scholar
  87. [87]
    E. Hewitt and K. A. Ross. Abstract Harmonic Analysis, volume I. Springer-Verlag, Berlin, Heidelberg, New York, 1963.Google Scholar
  88. [88]
    E. Hewitt and K. A. Ross. Abstract Harmonic Analysis, volume II. Springer-Verlag, Berlin, Heidelberg, New York, 1970.Google Scholar
  89. [89]
    E. Hewitt and H. S. Zuckerman. Some theorems on lacunary Fourier series, with extensions to compact groups. Trans. Amer. Math Soc., 93:1–19, 1959.Google Scholar
  90. [90]
    E. Hewitt and H. S. Zuckerman. Singular measures with absolutely continuous convolution squares. Proc. Cambridge Phil. Soc., 62:399–420, 1966.Google Scholar
  91. [91]
    E. Hewitt and H. S. Zuckerman. Singular measures with absolutely continuous convolution squares-corrigendum. Proc. Cambridge Phil. Soc., 63:367–368, 1967.Google Scholar
  92. [92]
    E. Hille. Analytic Function Theory, volume II. Chelsea, New York,, N. Y., 1973.Google Scholar
  93. [93]
    B. Host and F. Parreau. Ensembles de Rajchman et ensembles de continuité. C. R. Acad. Sci. Paris, 288:A899–A902, 1979.Google Scholar
  94. [94]
    B. Host, J.-F. Méla, and F. Parreau. Analyse Harmonique des Mesures, volume 135-136 of Astérisque. Soc. Math. France, Paris, 1986.Google Scholar
  95. [95]
    R. A. Hunt. On the convergence of Fourier series. In 1968 Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), pages 235–255. Southern Illinois Univ. Press, Carbondale, Ill., 1968.Google Scholar
  96. [96]
    M. F. Hutchinson. Non-tall compact groups admit infinite Sidon sets. J. Aust. Math. Soc., 23(4):467–475, 1977.Google Scholar
  97. [97]
    J. Johnsen. Simple proofs of nowhere-differentiability for Weierstrass’s function and cases of slow growth. J. Fourier Anal. Appl., 16(1):17–33, 2010.Google Scholar
  98. [98]
    G. W. Johnson and G. S. Woodward. On p-Sidon sets. Indiana Univ. Math J., 24:161–167, 1974/75.Google Scholar
  99. [99]
    J.-P. Kahane. Sur les fonctions moyennes-périodique bornées. Ann. Inst. Fourier (Grenoble), 7:293–314, 1957.Google Scholar
  100. [100]
    J.-P. Kahane. Ensembles de Ryll-Nardzewski et ensembles de Helson. Colloquium Math., 15:87–92, 1966.Google Scholar
  101. [101]
    J.-P. Kahane. Séries de Fourier Absolument Convergentes, volume 50 of Ergebnisse der Math. Springer, Berlin, Heidelberg, New York, 1970.Google Scholar
  102. [102]
    J.-P. Kahane. Algèbres tensorielles et analyse harmonique. In Séminaire Bourbaki, Années 1964/1965-1965/1966, Exposés 277-312, pages 221–230. Société Math. France, Paris, 1995.Google Scholar
  103. [103]
    J.-P. Kahane. Un théorème de Helson pour des séries de Walsh. In Linear and Complex Analysis, volume 226 of Amer. Math. Soc. Transl. Ser. 2, pages 67–73. Amer. Math. Soc., Providence, RI, 2009.Google Scholar
  104. [104]
    J.-P. Kahane and Y. Katznelson. Entiers aléatoires et analyse harmonique. J. Anal. Math., 105:363–378, 2008.Google Scholar
  105. [105]
    J.-P. Kahane and Y. Katznelson. Distribution uniforme de certaines suites d’entiers aléatoires dans le groupe de Bohr. J. Anal. Math., 105: 379–382, 2008.Google Scholar
  106. [106]
    J.-P. Kahane and R. Salem. Ensembles Parfaits et Séries Trigonométriques (Nouvelle Édition). Hermann, Paris, 1994.Google Scholar
  107. [107]
    N. J. Kalton. On vector-valued inequalities for Sidon sets and sets of interpolation. Colloq. Math. , 64(2):233–244, 1993.Google Scholar
  108. [108]
    Y. Katznelson. An Introduction to Harmonic Analysis. Cambridge Mathematical Library. Cambridge University Press, Cambridge, U. K., 3rd edition, 2004.Google Scholar
  109. [109]
    Y. Katznelson and P. Malliavin. Vérification statistique de la conjecture de la dichotomie sur une classe d’algèbres de restriction. C. R. Acad. Sci. Paris, 262:A490–A492, 1966.Google Scholar
  110. [110]
    R. Kaufman and N. Rickert. An inequality concerning measures. Bull. Amer. Math. Soc., 72(4):672–676, 1966.Google Scholar
  111. [111]
    A. Kechris and A. Louveau. Descriptive Set Theory and the Structure of Sets of Uniqueness. Number 128 in London Math. Soc. Lecture Notes. Cambridge Univ. Press, Cambridge, U. K., 1987.Google Scholar
  112. [112]
    J. H. B. Kemperman. On products of sets in a locally compact group. Fund. Math., 56:51–68, 1964.Google Scholar
  113. [113]
    S. V. Kislyakov. Banach spaces and classical harmonic analysis. In Handbook of the Geometry of Banach Spaces, volume I, pages 871–898. Elsivier, Amsterdam, London, New York, 2001.Google Scholar
  114. [114]
    M. Kneser. Summendmengen in lokalkompakten abelschen Gruppen. Math. Zeitschrift, 66:88–110, 1956.Google Scholar
  115. [115]
    A. Kolmogorov. Une contribution à l’étude de la convergence des séries de Fourier. Fund. Math., 5:96–97, 1924.Google Scholar
  116. [116]
    K. Kunen and W. Rudin. Lacunarity and the Bohr topology. Math. Proc. Cambridge Philos. Soc., 126:117–137, 1999.Google Scholar
  117. [117]
    S. Kwapień and A. Pełczyński. Absolutely summing operators and translation-invariant spaces of functions on compact abelian groups. Math. Nachr., 94:303–340, 1980.Google Scholar
  118. [118]
    N. Levinson. Gap and Density Theorems, volume 26 of A. M. S. Colloquium Publications. American Math. Soc., Providence, R. I., 1940.Google Scholar
  119. [119]
    D. Li and H. Queffélec. Introduction à l’Étude des Espaces de Banach, Analyse et Probabilités. Societé Mathématique de France, Paris, 2004.Google Scholar
  120. [120]
    L.-Å. Lindahl and F. Poulsen. Thin Sets in Harmonic Analysis. Dekker, New York, N. Y., 1971.Google Scholar
  121. [121]
    J. S. Lipiński. Sur un problème de E. Marczewski concernant des fonctions péroidiques. Bull. Acad. Pol. Sci,. Sér. Math., Astr. Phys., 8:695–697, 1960.Google Scholar
  122. [122]
    J. S. Lipiński. On periodic extensions of functions. Colloquium Math., 13:65–71, 1964.Google Scholar
  123. [123]
    J. López and K. A. Ross. Sidon Sets, volume 13 of Lecture Notes in Pure and Applied Math. Marcel Dekker, New York, N. Y., 1975.Google Scholar
  124. [124]
    F. Lust(-Piquard). Sur la réunion de deux ensembles de Helson. C. R. Acad. Sci. Paris, 272:A720–A723, 1971.Google Scholar
  125. [125]
    F. Lust(-Piquard). L’espace des fonctions presque-périodiques dont le spectre est contenu dans un ensemble compact dénombrable a la proprièté de Schur. Colloquium Math., 41:273–284, 1979.Google Scholar
  126. [126]
    M. P. Malliavin-Brameret and P. Malliavin. Caractérisation arithmétique des ensembles de Helson. C. R. Acad. Sci. Paris, 264: 192–193, 1967.Google Scholar
  127. [127]
    M. B. Marcus and G. Pisier. Random Fourier Series with Applications to Harmonic Analysis, volume 101 of Annals of Math. Studies. Princeton University Press, Princeton, N. J., 1981.Google Scholar
  128. [128]
    J.-F. Méla. Sur certains ensembles exceptionnels en analyse de Fourier. Ann. Inst. Fourier (Grenoble), 18:31–71, 1968.Google Scholar
  129. [129]
    J.-F. Méla. Sur les ensembles d’interpolation de C. Ryll-Nardzewski et de S. Hartman. Studia Math., 29:168–193, 1968.Google Scholar
  130. [130]
    J.-F. Méla. Approximation diophantienne et ensembles lacunaires. Mémoires Soc. Math. France, 19:26–54, 1969.Google Scholar
  131. [131]
    J.-F. Méla. Private communication, 2010.Google Scholar
  132. [132]
    Y. Meyer. Elargissement des ensembles de Sidon sur la droite. In Seminaire d’Analyse Harmonique Orsay, number 2 in Publ. Math. Univ. Paris VII, pages 1–14. Faculté des Sciences (Univ. Paris-Sud), Orsay, France, 1967/1968.
  133. [133]
    I. M. Miheev. Series with gaps. Mat. Sb., (N.S.) 98(140)(4(12)): 538–563, 639, 1975. in Russian.Google Scholar
  134. [134]
    I. M. Miheev. On lacunary series. Mat. Sb., 27(4):481–502, 1975. translation of ’Series with gaps’.Google Scholar
  135. [135]
    L. J. Mordell. On power series with circle of convergence as a line of essential singularities. J. London Math. Soc., 2:146–148, 1927.Google Scholar
  136. [136]
    S. A. Morris. Pontryagin Duality and the Structure of Locally Compact Abelian Groups. Number 29 in London Mathematical Society Lecture Notes. Cambridge University Press, Cambridge-New York-Melbourne, 1977.Google Scholar
  137. [137]
    J. Mycielski. On a problem of interpolation by periodic functions. Colloquium Math., 8:95–97, 1961.Google Scholar
  138. [138]
    K. O’Bryant. A complete annotated bibliography of work related to Sidon sequences. Electronic Journal of Combinatorics, DS11, 2004.Google Scholar
  139. [139]
    H. Okamoto. A remark on continuous, nowhere differentiable functions. Proc. Japan Acad., 81(3):47–50, 2005.Google Scholar
  140. [140]
    A. Pajor. Plongement de 1 n dans les espaces de Banach complexes. C. R. Acad. Sci. Paris, 296:741–743, 1983.
  141. [141]
    A. Pajor. Plongement de 1 K complexe dans les espaces de Banach. In Seminar on the geometry of Banach spaces, Vol. I, II (Paris, 1983), number 18 in Publ. Math. Univ. Paris VII, pages 139–148. Univ. Paris VII, Paris, 1984.
  142. [142]
    A. Pajor. Sous-espaces ℓ 1 n des Espaces de Banach. Number 16 in Travaux en Cours. Hermann, Paris, 1985. ISBN 2-7056-6021-6.
  143. [143]
    W. Parker. Central Sidon and central Λ p sets. J. Aust. Math. Soc, 14:62–74, 1972.Google Scholar
  144. [144]
    M. Pavlović. Lacunary series in weighted spaces of analytic functions. Archiv der Math., 97(5):467–473, 2011.Google Scholar
  145. [145]
    L. Pigno. Fourier-Stieltjes transforms which vanish at infinity off certain sets. Glasgow Math. J., 19:49–56, 1978.Google Scholar
  146. [146]
    G. Pisier. Ensembles de Sidon et espace de cotype 2. In Séminaire sur la Géométrie des Espaces de Banach, volume 14, Palaiseau, France, 1977-1978. École Polytech.
  147. [147]
    G. Pisier. Sur l’espace de Banach des séries de Fourier aléatoires presque sûrement continues. In Séminaire sur la Géométrie des Espaces de Banach, volume 17-18, Palaiseau, France, 1977-1978. École Polytech.
  148. [148]
    G. Pisier. Ensembles de Sidon et processus Gaussiens. C. R. Acad. Sci. Paris, 286(15):A671–A674, 1978.Google Scholar
  149. [149]
    G. Pisier. De nouvelles caractérisations des ensembles de Sidon. In Mathematical analysis and applications, Part B, volume 7b of Adv. in Math. Suppl. Studies, pages 685–726. Academic Press, New York, London, 1981.Google Scholar
  150. [150]
    G. Pisier. Conditions d’entropie et caractérisations arithmétique des ensembles de Sidon. In Proc. Conf. on Modern Topics in Harmonic Analysis, pages 911–941, Torino/Milano, 1982. Inst. de Alta Mathematica.Google Scholar
  151. [151]
    G. Pisier. Arithmetic characterizations of Sidon sets. Bull. Amer. Math. Soc., (N.S.) 8(1):87–89, 1983.Google Scholar
  152. [152]
    Ch. Pommerenke. Lacunary power series and univalent functions. Mich. Math. J., 11:219–223, 1964.Google Scholar
  153. [153]
    J. Price. Lie Groups and Compact Groups. Number 25 in London Math. Soc. Lecture Notes. Cambridge Univ. Press, Cambridge, U. K., 1977.Google Scholar
  154. [154]
    M. Queffélec. Substitution Dynamical Systems−Spectral Analysis, volume 1294 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 2nd edition, 2010.Google Scholar
  155. [155]
    D. Ragozin. Central measures on compact simple Lie groups. J. Func. Anal., 10:212–229, 1972.Google Scholar
  156. [156]
    (L.) T. Ramsey. A theorem of C. Ryll-Nardzewski and metrizable l.c.a. groups. Proc. Amer. Math. Soc., 78(2):221–224, 1980.Google Scholar
  157. [157]
    (L.) T. Ramsey. Bohr cluster points of Sidon sets. Colloquium Math., 68(2):285–290, 1995.Google Scholar
  158. [158]
    (L.) T. Ramsey. Comparisons of Sidon and I 0 sets. Colloquium Math., 70(1):103–132, 1996.Google Scholar
  159. [159]
    (L.) T. Ramsey and B. B. Wells. Interpolation sets in bounded groups. Houston J. Math., 10:117–125, 1984.Google Scholar
  160. [160]
    D. G. Rider. Gap series on groups and spheres. Canadian J. Math., 18:389–398, 1966.Google Scholar
  161. [161]
    D. G. Rider. Central lacunary sets. Monatsh. Math., 76:328–338, 1972.Google Scholar
  162. [162]
    D. G. Rider. Randomly continuous functions and Sidon sets. Duke Math. J., 42:759–764, 1975.Google Scholar
  163. [163]
    F. Riesz. Über die Fourierkoeffizienten einer stetigen Funktionen von beschränkter Schwankung. Math. Zeitschr., 18:312–315, 1918.Google Scholar
  164. [164]
    H. Rosenthal. On trigonometric series associated with weak ∗  closed subspaces of continuous functions. J. Math. Mech. (Indiana Univ. Math. J.), 17:485–490.Google Scholar
  165. [165]
    J. J. Rotman. An Introduction to the Theory of Groups, volume 148 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 4th edition, 1995 (2nd printing, 1999).Google Scholar
  166. [166]
    W. Rudin. Trigonometric series with gaps. J. Math. Mech. (Indiana Univ. Math. J.), 9(2):203–227, 1960.Google Scholar
  167. [167]
    W. Rudin. Fourier Analysis on Groups. Wiley Interscience, New York, N. Y., 1962.Google Scholar
  168. [168]
    C. Ryll-Nardzewski. Remarks on interpolation by periodic functions. Bull. Acad. Pol. Sci, Sér. Math., Astr., Phys., 11:363–366, 1963.Google Scholar
  169. [169]
    C. Ryll-Nardzewski. Concerning almost periodic extensions of functions. Colloquium Math., 12:235–237, 1964.Google Scholar
  170. [170]
    S. Saeki. On the union of two Helson sets. J. Math. Soc. Japan, 23: 636–648, 1971.Google Scholar
  171. [171]
    J. A. Seigner. Rademacher variables in connection with complex scalars. Acta Math. Univ. Comenianae, 66:329–336, 1997.Google Scholar
  172. [172]
    A. Shields. Sur la mesure d’une somme vectorielle. Fund. Math., 42: 57–60, 1955.Google Scholar
  173. [173]
    S. Sidon. Ein Satz uber die absolute Konvergenz von Fourierreihen in dem sehr viele Glieder fehlen. Math. Ann., 96:418–419, 1927.Google Scholar
  174. [174]
    S. Sidon. Veralgemeinerung eines Satzes über die absolute Konvergen von Fourierreihen mit Lücken. Math. Ann., 97:675–676, 1927.Google Scholar
  175. [175]
    B. P. Smith. Helson sets not containing the identity are uniform Fatou–Zygmund sets. Indiana Univ. Math. J., 27:331–347, 1978.Google Scholar
  176. [176]
    S. B. Stečkin. On the absolute convergence of Fourier series. Izv. Akad. Nauk. S.S.S.R., 20:385, 1956.Google Scholar
  177. [177]
    J. D. Stegeman. On union of Helson sets. Indag. Math., 32:456–462, 1970.Google Scholar
  178. [178]
    A. Stöhr. Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. I. J. Reine Angew. Math., 194:40–65, 1955.Google Scholar
  179. [179]
    A. Stöhr. Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. II. J. Reine Angew. Math., 194:111–140, 1955.Google Scholar
  180. [180]
    E. Strzelecki. On a problem of interpolation by periodic and almost periodic functions. Colloquium Math., 11:91–99, 1963.Google Scholar
  181. [181]
    E. Strzelecki. Some theorems on interpolation by periodic functions. Colloquium Math., 12:239–248, 1964.Google Scholar
  182. [182]
    E. Szemeredi. On sets of integers containing no k elements in arithmetic progression. Acta Arith., 27:199–245, 1975.Google Scholar
  183. [183]
    A. Ülger. An abstract form of a theorem of Helson and applications to sets of synthesis and sets of uniqueness. J. Functional Anal., 258(3): 956–977, 2010.Google Scholar
  184. [184]
    E. K. van Douwen. The maximal totally bounded group topology on G and the biggest minimal G-space, for abelian groups G. Topology Appl., 34(1):69–91, 1990.Google Scholar
  185. [185]
    V. S. Varadarajan. Lie groups and Lie algebras and their Representations. Number 102 in Graduate Texts in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 1984. ISBN 0-387-90969-9.Google Scholar
  186. [186]
    N. Th. Varopoulos. Sur les ensembles parfaits et les séries trigonométriques. C. R. Acad. Sci. Paris, 260:A3831–A3834, 1965.Google Scholar
  187. [187]
    N. Th. Varopoulos. Tensor algebras over discrete spaces. J. Functional Anal., 3:321–335, 1969.Google Scholar
  188. [188]
    N. Th. Varopoulos. Groups of continuous functions in harmonic analysis. Acta Math, 125:109–154, 1970.Google Scholar
  189. [189]
    N. Th. Varopoulos. Sur la réunion de deux ensembles de Helson. C. R. Acad. Sci. Paris, 271:A251–A253, 1970.Google Scholar
  190. [190]
    N. Th. Varopoulos. Une remarque sur les ensembles de Helson. Duke Math. J., 43:387–390, 1976.Google Scholar
  191. [191]
    R. Vrem. Independent sets and lacunarity for hypergroups. J. Austral. Math. Soc., 50(2):171–188, 1991.Google Scholar
  192. [192]
    K. Weierstrass. Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen. In Mathematische Werke von Karl Weierstrass, Vol. 2, pages 71–74. Meyer, Berlin, 1895.Google Scholar
  193. [193]
    G. Weiss and M. Weiss. On the Picard property of lacunary power series. Studia Math., 22:221–245, 1963.Google Scholar
  194. [194]
    M. Weiss. Concerning a theorem of Paley on lacunary power series. Acta Math., 102:225–238, 1959.Google Scholar
  195. [195]
    B. Wells. Restrictions of Fourier transforms of continuous measures. Proc. Amer. Math. Soc., 38:92–94, 1973.Google Scholar
  196. [196]
    D. C. Wilson. On the structure of Sidon sets. Monatsh. für Math., 101:67–74, 1986.Google Scholar
  197. [197]
    A. Zygmund. On the convergence of lacunary trigonometric series. Fund. Math., 16:90–107, 1930.Google Scholar
  198. [198]
    A. Zygmund. Sur les séries trigonométriques lacunaires. J. London Math. Soc., 18:138–145, 1930.Google Scholar
  199. [199]
    A. Zygmund. Trigonometric Series, volume I. Cambridge University Press, Cambridge, U. K., 2 edition, 1959.Google Scholar
  200. [200]
    A. Zygmund. Trigonometric Series, volume II. Cambridge University Press, Cambridge, U. K., 2 edition, 1959.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Colin C. Graham
    • 1
  • Kathryn E. Hare
    • 2
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations