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Thomas-William Korner
Some results on Kronecker, Dirichlet and Helson sets
Annales de l'institut Fourier, 20 no. 2 (1970), p. 219-324, doi: 10.5802/aif.355
Article PDF | Reviews MR 44 #1995 | Zbl 0196.08403 | 2 citations in Cedram

Résumé - Abstract

We construct the following: a perfect non Dirichlet set every proper closed subset of which is Kronecker, A weak Kronecker set which is not an $R$ set; an independent countable Dirichlet set which is not Kronecker; a collection of $q$-disjoint Kronecker sets whose union is independent but Helson $1/q$; A countable collection of disjoint Kronecker sets whose union is closed and independent but not Helson: a perfect independent Dirichlet set which is not Helson.

Bibliography

[1] J. ARBAULT, Sur l'ensemble de convergence absolue d'une série trigonométrique, Bull. Soc. Math. France, t. 80, (1952), 254-317. Numdam |  MR 14,1080d |  Zbl 0048.04202
[2] N. K. BARY, A Treatise on Trigonometric Series, Vol. II., English translation Pergamon Press, Oxford (1964).  MR 30 #1347 |  Zbl 0129.28002
[3] J. W. S. CASSELS, An Introduction to Diophantine Approximation, Cambridge University Press, (1965).
[4] G. H. HARDY and E. M. WRIGHT, Introduction to the Theory of Numbers, 4th edition, Oxford University Press, (1959).
[5] S. HARTMAN and C. RYLL-NARDZEWSKI, Über die Spaltung von Fourierreihen fast periodischer Funktionen, Studia Mathematica 19, (1960) 287-295. Article |  MR 22 #9805 |  Zbl 0093.08301
[6] F. HAUDSORFF, Set Theory, English translation, Chelsea, New York (1957).  Zbl 0081.04601
[7] E. HEWITT and K. A. ROSS, Abstract Harmonic Analysis, Springer Verlag, Berlin (1963).
[8] J.-P. KAHANE, Approximation par des exponentielles imaginaires ; ensembles de Dirichlet et ensembles de Kronecker. Journal of Approximation Theory 2, (1969).
[9] J.-P. KAHANE and R. SALEM, Ensembles Parfaits et Séries Trigonométriques, Hermann, Paris (1963).  MR 28 #3279 |  Zbl 0112.29304
[10] Y. KATZNELSON, An Introduction to Harmonic Analysis. John Wiley and Sons, New York, (1968).  MR 40 #1734 |  Zbl 0169.17902
[11] R. KAUFMAN, A Functional Method for Linear Sets, Israel J. Math. 5, (1967).  MR 38 #4902 |  Zbl 0156.37403
[12] O. ORE, Theory of Graphs, American Mathematical Society Colloquium Publications, Vol. XXXVIII, A.M.S.: Rhode Island (1962).  MR 27 #740 |  Zbl 0105.35401
[13] W. RUDIN, Fourier Analysis on Groups, John Wiley and Sons, New York, (1967).
[14] R. SALEM, Œuvres Mathématiques, Hermann, Paris (1967).  MR 36 #22 |  Zbl 0145.06901
[15] I. WIK, Some Examples of Sets with Linear Independence, Ark. Mat. 5, (1965), 207-214.  Zbl 0127.29403
[16] A. BERNARD and N.-Th. VAROPOULOS, Groupes des fonctions continues sur un compact, Studia Mathematica 35, (1970), 199-205. Article |  Zbl 0199.46602
[17] N.-Th. VAROPOULOS, Groups of Continuous Functions in Harmonic Analysis Acta. Math, 125, (1970), 109-154.  MR 43 #7868 |  Zbl 0214.38102
[18] N.-Th. VAROPOULOS, Comptes Rendus Acad. Sci., Paris, 268, (1969), 954-957.  Zbl 0187.07301
[19] N.-Th. VAROPOULOS, A problem on Kronecker sets, Studia Mathematica 37, (1970), 95-101. Article |  MR 42 #3493 |  Zbl 0202.14002
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