Abstract
I 0 sets characterized analytically, in terms of function algebras, and topologically. I 0 sets do not cluster at continuous characters.
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We a redropp i n g the “E ” f romtheAP(… ) not a t i on, a s f orew a rned.
References
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.
M. E. Andersson. The Kaufman-Rickert inequality governs Rademacher sums. Analysis (Munich), 23:65–79, 2003.
J. Arias de Reyna. Pointwise Convergence of Fourier Series, volume 1785 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 2002.
N. Asmar and S. Montgomery-Smith. On the distribution of Sidon series. Arkiv för Math., 31(1):13–26, 1993.
G. Bachelis and S. Ebenstein. On Λ(p) sets. Pacific J. Math., 54:35–38, 1974.
S. Banach. Über einige Eigenschaften der lakunäre trigonometrischen Reihen, II. Studia Math., 2:207–220, 1930.
N. K. Bary. A Treatise on Trigonometric Series, volume I. MacMillan, New York, N. Y., 1964.
F. Baur. Lacunarity on nonabelian groups and summing operators. J. Aust. Math. Soc., 71(1):71–79, 2001.
G. Benke. Arithmetic structure and lacunary Fourier series. Proc. Amer. Math. Soc., 34:128–132, 1972.
G. Benke. On the hypergroup structure of central Λ(p) sets. Pacific J. Math., 50:19–27, 1974.
D. Berend. Parallelepipeds in sets of integers. J. Combin. Theory, 45 (2):163–170, 1987.
K. G. Binmore. Analytic functions with Hadamard gaps. Bull. London Math. Soc., 1:211–217, 1969.
R. C. Blei. On trigonometric, series associated with separable, translation invariant subspaces of L ∞. Trans. Amer. Math. Soc., 173:491–499, 1972.
R. C. Blei. Fractional Cartesian products of sets. Ann. Inst. Fourier (Grenoble), 29(2):79–105, 1979.
S. Bochner. Monotone Funktionen, Stieltjessche Integrale, und harmonische Analyse. Math. Ann., 108:378–410, 1933.
A. Bonami. Étude des coefficients de Fourier des fonctions de L p(G). Ann. Inst. Fourier (Grenoble), 20(2):335–402, 1970.
J. Bourgain. Propriétés de décomposition pour les ensembles de Sidon. Bull. Soc. Math. France, 111(4):421–428, 1983.
J. Bourgain. Subspaces of ℓ 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.
J. Bourgain. Sidon sets and Riesz products. Ann. Inst. Fourier (Grenoble), 35(1):137–148, 1985.
J. Bourgain. A remark on entropy of abelian groups and the invariant uniform approximation property. Studia Math., 86:79–84, 1987.
J. Bourgan and V. Milḿan. Dichotomie du cotype pours les espace invariants. C. R. Acad. Sci. Paris, pages 435–438, 1985.
L Carleson. On convergence and growth of partial sums of Fourier series. Acta Math., 116:135–157, 1966.
D. Cartwright and J. McMullen. A structural criterion for Sidon sets. Pacific J. Math., 96:301–317, 1981.
Mei-Chu Chang. On problems of Erdös and Rudin. J. Functional Anal., 207(2):444–460, 2004.
Y. S. Chow and H. Teicher. Probability Theory. Independence, Interchangeability, Martingales. Springer Texts in Statistics. Springer-Verlag, Berlin, Heidelberg, New York, 3rd edition, 1997.
E. Crevier. Private communication. 2012.
M. Déchamps(-Gondim). Ensembles de Sidon topologiques. Ann. Inst. Fourier (Grenoble), 22(3):51–79, 1972.
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.
M. Déchamps(-Gondim). Densité harmonique et espaces de Banach invariants par translation ne contenant pas c 0. Colloquium Math., 51:67–84, 1987.
M. Déchamps(-Gondim) and O. Selles. Compacts associés aus sommes de suites lacunaires. Publ. Math. Orsay, 1:27–40, 1996.
R. Doss. Elementary proof of a theorem of Helson. Proc. Amer. Math. Soc., 27(2):418–420, 1971.
S. W. Drury. Sur les ensembles de Sidon. C. R. Acad. Sci. Paris, 271A: 162–163, 1970.
S. W. Drury. The Fatou-Zygmund property for Sidon sets. Bull. Amer. Math. Soc., 80:535–538, 1974.
W. F. Eberlein. Characterizations of Fourier-Stieltjes transforms. Duke Math. J., 22:465–468, 1955.
R. E. Edwards and K. A. Ross. p-Sidon sets. J. Functional Anal., 15:404–427, 1974.
R. E. Edwards, E. Hewitt, and K. A. Ross. Lacunarity for compact groups, III. Studia Math., 44:429–476, 1972.
W. Feller. An Introduction to Probability Theory and its Applications. John Wiley and Sons, New York, London, 2nd edition, 1957.
A. Figá-Talamanca and D. G. Rider. A theorem of Littlewood and lacunary series for compact groups. Pacific J. Math., 16:505–514, 1966.
John J. F. Fournier and Louis Pigno. Analytic and arithmetic properties of thin sets. Pacific J. Math., 105(1):115–141, 1983.
W. H. J. Fuchs. On the zeros of a power series with Hadamard gaps. Nagoya Math. J., 29:167–174, 1967.
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.
J. Galindo and S. Hernández. Interpolation sets and the Bohr topology of locally compact groups. Adv. in Math., 188:51–68, 2004.
I. M. Gel’fand. Normierte Ringe. Mat. Sb., N.S., 9:3–24, 1941.
J. Gerver. The differentiability of the Riemann function at certain rational multiples of π. Amer. J. Math., 92:33–55, 1970.
J. Gerver. More on the differentiability of the Riemann function. Amer. J. Math., 93:33–41, 1971.
B. N. Givens and K. Kunen. Chromatic numbers and Bohr topologies. Topology Appl., 131(2):189–202, 2003.
D. Gnuschke and Ch. Pommerenke. On the radial limits of functions with Hadamard gaps. Michigan Math. J., 32(1):21–31, 1985.
E. Goursat. A Course in Mathematical Analysis, volume I. Ginn & Co., Boston, Chicago, London, New York, 1904. E. R. Hedrick, trans.
W. T. Gowers. A new proof of Szemeredi’s theorem. Geom. Functional Anal., 11:465–588, 2001.
C. C. Graham. Sur un théorème de Katznelson et McGehee. C. R. Acad. Sci. Paris, 276:A37–A40, 1973.
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.
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.
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.
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.
C. C. Graham and K. E. Hare. Characterizing Sidon sets by interpolation properties of subsets. Colloquium Math., 112(2):175–199, 2008.
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.
C. C. Graham and K. E. Hare. Characterizations of some classes of I 0 sets. Rocky Mountain J. Math., 40(2):513–525, 2010.
C. C. Graham and K. E. Hare. Sets of zero discrete harmonic density. Math. Proc. Cambridge Philos. Soc., 148(2):253–266, 2010.
C. C. Graham and K. E. Hare. Existence of large ε-Kronecker and FZI0(U) sets in discrete abelian groups. Colloquium Math., 2012.
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.
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.
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.
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.
D. Grow. A class of I 0-sets. Colloquium Math., 53(1):111–124, 1987.
D. Grow. Sidon sets and I 0-sets. Colloquium Math., 53(2):269–270, 1987.
D. Grow. A further note on a class of I 0-sets. Colloquium Math., 53(1):125–128, 1987.
D. Grow and K. E. Hare. The independence of characters on non-abelian groups. Proc. Amer. Math. Soc., 132(12):3641–3651, 2004.
D. Grow and K. E. Hare. Central interpolation sets for compact groups and hypergroups. Glasgow Math. J., 51(3):593–603, 2009.
A. Gut. An Intermediate Course in Probability. Springer Texts in Statistics. Springer, Berlin, Heidelberg, New York, 2nd edition, 2009.
J. Hadamard. Essai sur l’étude des fonctions données par leur développement de Taylor. J. Math. Pures Appl., 8(4):101–186, 1892.
A. Harcharras. Fourier analysis, Schur multipliers on S p and non-commutative Λ(p)-sets. Studia Math., 137(3):203–260, 1999.
G. H. Hardy. Weierstrass’s non-differentiable function. Trans. Amer. Math. Soc., 17:301–325, 1916.
K. E. Hare. Arithmetic properties of thin sets. Pacific J. Math., 131: 143–155, 1988.
K. E. Hare. An elementary proof of a result on Λ(p) sets. Proc. Amer. Math. Soc., 104:829–832, 1988.
K. E. Hare and (L.) T. Ramsey. I 0 sets in non-abelian groups. Math. Proc. Cambridge Philos. Soc., 135: 81–98, 2003.
K. E. Hare and (L.) T. Ramsey. Kronecker constants for finite subsets of integers. J. Fourier Anal. Appl., 18(2):326–366, 2012.
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.
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.
S. Hartman. On interpolation by almost periodic functions. Colloquium Math, 8:99–101, 1961.
S. Hartman. Interpolation par les mesures diffuses. Colloquium Math., 26:339–343, 1972.
S. Hartman and C. Ryll-Nardzewski. Almost periodic extensions of functions. Colloquium Math., 12:23–39, 1964.
S. Hartman and C. Ryll-Nardzewski. Almost periodic extensions of functions, II. Colloquium Math., 15:79–86, 1966.
S. Hartman and C. Ryll-Nardzewski. Almost periodic extensions of functions, III. Colloquium Math., 16:223–224, 1967.
H. Helson. Fourier transforms on perfect sets. Studia. Math., 14:209–213, 1954.
H. Helson and J.-P. Kahane. A Fourier method in diophantine problems. J. d’Analyse Math., 15:245–262, 1965.
C. Herz. Drury’s lemma and Helson sets. Studia Math., 42:205–219, 1972.
E. Hewitt and K. A. Ross. Abstract Harmonic Analysis, volume I. Springer-Verlag, Berlin, Heidelberg, New York, 1963.
E. Hewitt and K. A. Ross. Abstract Harmonic Analysis, volume II. Springer-Verlag, Berlin, Heidelberg, New York, 1970.
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.
E. Hewitt and H. S. Zuckerman. Singular measures with absolutely continuous convolution squares. Proc. Cambridge Phil. Soc., 62:399–420, 1966.
E. Hewitt and H. S. Zuckerman. Singular measures with absolutely continuous convolution squares-corrigendum. Proc. Cambridge Phil. Soc., 63:367–368, 1967.
E. Hille. Analytic Function Theory, volume II. Chelsea, New York,, N. Y., 1973.
B. Host and F. Parreau. Ensembles de Rajchman et ensembles de continuité. C. R. Acad. Sci. Paris, 288:A899–A902, 1979.
B. Host, J.-F. Méla, and F. Parreau. Analyse Harmonique des Mesures, volume 135-136 of Astérisque. Soc. Math. France, Paris, 1986.
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.
M. F. Hutchinson. Non-tall compact groups admit infinite Sidon sets. J. Aust. Math. Soc., 23(4):467–475, 1977.
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.
G. W. Johnson and G. S. Woodward. On p-Sidon sets. Indiana Univ. Math J., 24:161–167, 1974/75.
J.-P. Kahane. Sur les fonctions moyennes-périodique bornées. Ann. Inst. Fourier (Grenoble), 7:293–314, 1957.
J.-P. Kahane. Ensembles de Ryll-Nardzewski et ensembles de Helson. Colloquium Math., 15:87–92, 1966.
J.-P. Kahane. Séries de Fourier Absolument Convergentes, volume 50 of Ergebnisse der Math. Springer, Berlin, Heidelberg, New York, 1970.
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.
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.
J.-P. Kahane and Y. Katznelson. Entiers aléatoires et analyse harmonique. J. Anal. Math., 105:363–378, 2008.
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.
J.-P. Kahane and R. Salem. Ensembles Parfaits et Séries Trigonométriques (Nouvelle Édition). Hermann, Paris, 1994.
N. J. Kalton. On vector-valued inequalities for Sidon sets and sets of interpolation. Colloq. Math. , 64(2):233–244, 1993.
Y. Katznelson. An Introduction to Harmonic Analysis. Cambridge Mathematical Library. Cambridge University Press, Cambridge, U. K., 3rd edition, 2004.
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.
R. Kaufman and N. Rickert. An inequality concerning measures. Bull. Amer. Math. Soc., 72(4):672–676, 1966.
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.
J. H. B. Kemperman. On products of sets in a locally compact group. Fund. Math., 56:51–68, 1964.
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.
M. Kneser. Summendmengen in lokalkompakten abelschen Gruppen. Math. Zeitschrift, 66:88–110, 1956.
A. Kolmogorov. Une contribution à l’étude de la convergence des séries de Fourier. Fund. Math., 5:96–97, 1924.
K. Kunen and W. Rudin. Lacunarity and the Bohr topology. Math. Proc. Cambridge Philos. Soc., 126:117–137, 1999.
S. Kwapień and A. Pełczyński. Absolutely summing operators and translation-invariant spaces of functions on compact abelian groups. Math. Nachr., 94:303–340, 1980.
N. Levinson. Gap and Density Theorems, volume 26 of A. M. S. Colloquium Publications. American Math. Soc., Providence, R. I., 1940.
D. Li and H. Queffélec. Introduction à l’Étude des Espaces de Banach, Analyse et Probabilités. Societé Mathématique de France, Paris, 2004.
L.-Å. Lindahl and F. Poulsen. Thin Sets in Harmonic Analysis. Dekker, New York, N. Y., 1971.
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.
J. S. Lipiński. On periodic extensions of functions. Colloquium Math., 13:65–71, 1964.
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.
F. Lust(-Piquard). Sur la réunion de deux ensembles de Helson. C. R. Acad. Sci. Paris, 272:A720–A723, 1971.
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.
M. P. Malliavin-Brameret and P. Malliavin. Caractérisation arithmétique des ensembles de Helson. C. R. Acad. Sci. Paris, 264: 192–193, 1967.
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.
J.-F. Méla. Sur certains ensembles exceptionnels en analyse de Fourier. Ann. Inst. Fourier (Grenoble), 18:31–71, 1968.
J.-F. Méla. Sur les ensembles d’interpolation de C. Ryll-Nardzewski et de S. Hartman. Studia Math., 29:168–193, 1968.
J.-F. Méla. Approximation diophantienne et ensembles lacunaires. Mémoires Soc. Math. France, 19:26–54, 1969.
J.-F. Méla. Private communication, 2010.
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. http://www.math.u-psud.fr/~bib-lio/numer-isation/docs/Seminaire_d_analyse_harmoniquedOrsay_1967-1968/pdf/Seminaire_d_analyse_harmonique_d_Or-say_1967-1968.pdf.
I. M. Miheev. Series with gaps. Mat. Sb., (N.S.) 98(140)(4(12)): 538–563, 639, 1975. in Russian.
I. M. Miheev. On lacunary series. Mat. Sb., 27(4):481–502, 1975. translation of ’Series with gaps’.
L. J. Mordell. On power series with circle of convergence as a line of essential singularities. J. London Math. Soc., 2:146–148, 1927.
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.
J. Mycielski. On a problem of interpolation by periodic functions. Colloquium Math., 8:95–97, 1961.
K. O’Bryant. A complete annotated bibliography of work related to Sidon sequences. Electronic Journal of Combinatorics, DS11, 2004.
H. Okamoto. A remark on continuous, nowhere differentiable functions. Proc. Japan Acad., 81(3):47–50, 2005.
A. Pajor. Plongement de ℓ 1 n dans les espaces de Banach complexes. C. R. Acad. Sci. Paris, 296:741–743, 1983. http://perso-math.univ-mlv.fr/users/pajor.alain/recherche/pub.htm.
A. Pajor. Plongement de ℓ 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. http://perso-math.univ-mlv.fr/users/pajor.alain/recherche/pub.htm.
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. http://perso-math.univ-mlv.fr/users/pajor.alain/recherche/pub.htm.
W. Parker. Central Sidon and central Λ p sets. J. Aust. Math. Soc, 14:62–74, 1972.
M. Pavlović. Lacunary series in weighted spaces of analytic functions. Archiv der Math., 97(5):467–473, 2011.
L. Pigno. Fourier-Stieltjes transforms which vanish at infinity off certain sets. Glasgow Math. J., 19:49–56, 1978.
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. http://archive.numdam.org/ARCHIVE/SAF/SAF_1977-1978___/SAF_1977-1978____A11_0/SAF_1977-1978____A11_0.pdf.
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. http://archive.numdam.org/ARCHIVE/SAF/SAF_1977-1978___/SAF_1977-1978____A13_0/SAF_1977-1978____A13_0.pdf.
G. Pisier. Ensembles de Sidon et processus Gaussiens. C. R. Acad. Sci. Paris, 286(15):A671–A674, 1978.
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.
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.
G. Pisier. Arithmetic characterizations of Sidon sets. Bull. Amer. Math. Soc., (N.S.) 8(1):87–89, 1983.
Ch. Pommerenke. Lacunary power series and univalent functions. Mich. Math. J., 11:219–223, 1964.
J. Price. Lie Groups and Compact Groups. Number 25 in London Math. Soc. Lecture Notes. Cambridge Univ. Press, Cambridge, U. K., 1977.
M. Queffélec. Substitution Dynamical Systems−Spectral Analysis, volume 1294 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 2nd edition, 2010.
D. Ragozin. Central measures on compact simple Lie groups. J. Func. Anal., 10:212–229, 1972.
(L.) T. Ramsey. A theorem of C. Ryll-Nardzewski and metrizable l.c.a. groups. Proc. Amer. Math. Soc., 78(2):221–224, 1980.
(L.) T. Ramsey. Bohr cluster points of Sidon sets. Colloquium Math., 68(2):285–290, 1995.
(L.) T. Ramsey. Comparisons of Sidon and I 0 sets. Colloquium Math., 70(1):103–132, 1996.
(L.) T. Ramsey and B. B. Wells. Interpolation sets in bounded groups. Houston J. Math., 10:117–125, 1984.
D. G. Rider. Gap series on groups and spheres. Canadian J. Math., 18:389–398, 1966.
D. G. Rider. Central lacunary sets. Monatsh. Math., 76:328–338, 1972.
D. G. Rider. Randomly continuous functions and Sidon sets. Duke Math. J., 42:759–764, 1975.
F. Riesz. Über die Fourierkoeffizienten einer stetigen Funktionen von beschränkter Schwankung. Math. Zeitschr., 18:312–315, 1918.
H. Rosenthal. On trigonometric series associated with weak ∗ closed subspaces of continuous functions. J. Math. Mech. (Indiana Univ. Math. J.), 17:485–490.
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).
W. Rudin. Trigonometric series with gaps. J. Math. Mech. (Indiana Univ. Math. J.), 9(2):203–227, 1960.
W. Rudin. Fourier Analysis on Groups. Wiley Interscience, New York, N. Y., 1962.
C. Ryll-Nardzewski. Remarks on interpolation by periodic functions. Bull. Acad. Pol. Sci, Sér. Math., Astr., Phys., 11:363–366, 1963.
C. Ryll-Nardzewski. Concerning almost periodic extensions of functions. Colloquium Math., 12:235–237, 1964.
S. Saeki. On the union of two Helson sets. J. Math. Soc. Japan, 23: 636–648, 1971.
J. A. Seigner. Rademacher variables in connection with complex scalars. Acta Math. Univ. Comenianae, 66:329–336, 1997.
A. Shields. Sur la mesure d’une somme vectorielle. Fund. Math., 42: 57–60, 1955.
S. Sidon. Ein Satz uber die absolute Konvergenz von Fourierreihen in dem sehr viele Glieder fehlen. Math. Ann., 96:418–419, 1927.
S. Sidon. Veralgemeinerung eines Satzes über die absolute Konvergen von Fourierreihen mit Lücken. Math. Ann., 97:675–676, 1927.
B. P. Smith. Helson sets not containing the identity are uniform Fatou–Zygmund sets. Indiana Univ. Math. J., 27:331–347, 1978.
S. B. Stečkin. On the absolute convergence of Fourier series. Izv. Akad. Nauk. S.S.S.R., 20:385, 1956.
J. D. Stegeman. On union of Helson sets. Indag. Math., 32:456–462, 1970.
A. Stöhr. Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. I. J. Reine Angew. Math., 194:40–65, 1955.
A. Stöhr. Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. II. J. Reine Angew. Math., 194:111–140, 1955.
E. Strzelecki. On a problem of interpolation by periodic and almost periodic functions. Colloquium Math., 11:91–99, 1963.
E. Strzelecki. Some theorems on interpolation by periodic functions. Colloquium Math., 12:239–248, 1964.
E. Szemeredi. On sets of integers containing no k elements in arithmetic progression. Acta Arith., 27:199–245, 1975.
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.
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.
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.
N. Th. Varopoulos. Sur les ensembles parfaits et les séries trigonométriques. C. R. Acad. Sci. Paris, 260:A3831–A3834, 1965.
N. Th. Varopoulos. Tensor algebras over discrete spaces. J. Functional Anal., 3:321–335, 1969.
N. Th. Varopoulos. Groups of continuous functions in harmonic analysis. Acta Math, 125:109–154, 1970.
N. Th. Varopoulos. Sur la réunion de deux ensembles de Helson. C. R. Acad. Sci. Paris, 271:A251–A253, 1970.
N. Th. Varopoulos. Une remarque sur les ensembles de Helson. Duke Math. J., 43:387–390, 1976.
R. Vrem. Independent sets and lacunarity for hypergroups. J. Austral. Math. Soc., 50(2):171–188, 1991.
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.
G. Weiss and M. Weiss. On the Picard property of lacunary power series. Studia Math., 22:221–245, 1963.
M. Weiss. Concerning a theorem of Paley on lacunary power series. Acta Math., 102:225–238, 1959.
B. Wells. Restrictions of Fourier transforms of continuous measures. Proc. Amer. Math. Soc., 38:92–94, 1973.
D. C. Wilson. On the structure of Sidon sets. Monatsh. für Math., 101:67–74, 1986.
A. Zygmund. On the convergence of lacunary trigonometric series. Fund. Math., 16:90–107, 1930.
A. Zygmund. Sur les séries trigonométriques lacunaires. J. London Math. Soc., 18:138–145, 1930.
A. Zygmund. Trigonometric Series, volume I. Cambridge University Press, Cambridge, U. K., 2 edition, 1959.
A. Zygmund. Trigonometric Series, volume II. Cambridge University Press, Cambridge, U. K., 2 edition, 1959.
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Graham, C.C., Hare, K.E. (2013). I 0 Sets and Their Characterizations. In: Interpolation and Sidon Sets for Compact Groups. CMS Books in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5392-5_3
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