Hare, Kathryn E.
Central sidonicity for compact Lie groups
Annales de l'institut Fourier, Tome 45 (1995) no. 2 , p. 547-564
Zbl 0820.43003 | MR 96i:43004
doi : 10.5802/aif.1464
URL stable : http://www.numdam.org/item?id=AIF_1995__45_2_547_0

Soit G un groupe compact, connexe et non-abélien. Il est bien connu que le groupe dual G ^ peut ne pas contenir des sous-ensembles de type Sidon, infinis et centraux, mais on y trouve toujours, pour chaque p>1, des sous-ensembles de type p-Sidon qui sont aussi infinis et centraux. On montre, par une méthode essentiellement constructive, que les ensembles infinis centraux de type p-Sidon se trouvent aussi dans chaque sous-ensemble infini de G ^. Aussi étudions-nous, pour un groupe de Lie compact, la connexion entre sa sidonicité centrale et la sidonicité de son tore.
It is known that the dual of a compact, connected, non-abelian group may contain no infinite central Sidon sets, but always does contain infinite central p-Sidon sets for p>1. We prove, by an essentially constructive method, that the latter assertion is also true for every infinite subset of the dual. In addition, we investigate the relationship between weighted central Sidonicity for a compact Lie group and Sidonicity for its torus.


[1] D.I. Cartwright and J.R. Mcmullen, A structural criterion for the existence of infinite Sidon sets, Pacific J. Math., 96 (1981), 301-317. MR 83c:43009 | Zbl 0445.43006

[2] R. Coifman and G. Weiss, Central multiplier theorems for compact Lie groups, Bull. Amer. Math. Soc., 80 (1974), 124-126. MR 48 #9271 | Zbl 0276.43009

[3] A.H. Dooley, Central lacunary sets for Lie groups, J. Aust. Math. Soc., 45 (1988), 30-45. MR 89j:43007 | Zbl 0689.43003

[4] P. Gallagher, Zeroes of group characters, Math. Z., 87 (1965), 363-364. MR 31 #276 | Zbl 0128.25602

[5] K. Hare and D. Wilson, Weighted p-Sidon sets, J. Aust. Math. Soc., to appear. Zbl 0874.43005

[6] E. Hewitt and K. Ross, Abstract harmonic analysis II, Springer-Verlag, New York, 1970. Zbl 0213.40103

[7] J.E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972. MR 48 #2197 | Zbl 0254.17004

[8] J. Lopez and K. Ross, Sidon sets, Lecture Notes Pure Appl. Math., No. 13, Marcel Dekker, New York, 1975. MR 55 #13173 | Zbl 0351.43008

[9] W.A. Parker, Central Sidon and central Λ(p) sets, J. Aust. Math. Soc., 14 (1972), 62-74. MR 47 #9178 | Zbl 0237.43004

[10] J.F. Price, Lie groups and compact groups, London Math. Soc. Lecture Note Series No.25, Cambridge Univ. Press, Cambridge, 1977. MR 56 #8743 | Zbl 0348.22001

[11] D.L. Ragozin, Central measures on compact simple Lie groups, J. Func. Anal., 10 (1972), 212-229. MR 49 #5715 | Zbl 0286.43002

[12] D. Rider, Central lacunary sets, Monatsh. Math., 76 (1972), 328-338. MR 51 #3801 | Zbl 0258.43008

[13] R. Stanton and P. Tomas, Polyhedral summability of Fourier series on compact Lie groups, Amer. J. Math., 100 (1978), 477-493. MR 58 #29855 | Zbl 0421.43009

[14] V.S. Varadarajan, Lie groups, Lie algebras and their representations, Springer-Verlag, New York, 1984. Zbl 0955.22500