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I0 SETS FOR COMPACT, CONNECTED GROUPS: INTERPOLATION WITH MEASURES THAT ARE NONNEGATIVE OR OF SMALL SUPPORT

Part of: Abstract harmonic analysis Harmonic analysis in one variable

Published online by Cambridge University Press:  01 April 2008

COLIN C. GRAHAM  and
KATHRYN E. HARE
COLIN C. GRAHAM*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada (email: [email protected]) Mailing address: RR#1–D-156, Bowen Island, BC, V0N 1G0, Canada
KATHRYN E. HARE
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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In the dual object of an infinite compact, connected group, every infinite Sidon set contains an infinite subset on which full interpolation can be performed using very small classes of measures (discrete measures on arbitrarily small sets or nonnegative discrete measures). In particular, the Figà-Talamanca–Rider subset of an infinite product of compact, connected, simple Lie groups has these kinds of interpolation. This substantially improves previous interpolation results.

Keywords

compact nonabelian groupFatou–Zygmund setsFTR setI setSidon set

MSC classification

Secondary: 42A55: Lacunary series of trigonometric and other functions; Riesz products 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) 43A05: Measures on groups and semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 2 , April 2008 , pp. 199 - 215
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

The research of the authors is partially supported by NSERC.

References

[1]Cartwright, D. and McMullen, J., ‘A structural criterion for the existence of infinite Sidon sets’, Pacific. J. Math. 96 (1981), 301–317.CrossRefGoogle Scholar
[2]Déchamps-Gondim, M., ‘Ensembles de Sidon topologiques’, Ann. Inst. Fourier (Grenoble) 22 (1972), 51–79.CrossRefGoogle Scholar
[3]Graham, C. C. and Hare, K. E., ‘ε-Kronecker and I 0 sets in abelian groups, I: arithmetic properties of ε-Kronecker sets’, Math. Proc. Cambridge Philos. Soc. 140(3) (2006), 475–489.CrossRefGoogle Scholar
[4]Graham, C. C. and Hare, K. E., ‘ε-Kronecker and I 0 sets in abelian groups, III: Interpolation by measures on small sets’, Studia Math. 171(1) (2005), 15–32.Google Scholar
[5]Graham, C. C. and Hare, K. E., ‘ε-Kronecker and I 0 sets in abelian groups, IV: Interpolation by nonnegative measures’, Studia Math. 177(1) (2006), 9–24.Google Scholar
[6]Hare, K. E., ‘Central Sidonicity for compact Lie groups’, Ann. Inst. Fourier (Grenoble) 45 (1995), 547–564.CrossRefGoogle Scholar
[7]Hare, K. E. and Ramsey, L. T., ‘I 0 sets in nonabelian groups’, Math. Proc. Cambridge Philos. Soc. 135 (2003), 81–98.Google Scholar
[8]Kemperman, J. H. B., ‘On products of sets in a locally compact group’, Fund. Math. 56 (1964), 51–68.Google Scholar
[9]Lopez, J. and Ross, K., Sidon Sets, Lecture Notes in Pure and Applied Math., 13 (Marcel Dekker, New York, 1975).Google Scholar
[10]Price, J., Lie Groups and Compact Groups, London Math. Soc. Lecture Note Series No. 25 (Cambridge University Press, Cambridge, 1977).Google Scholar
[11]Varadarajan, V. S., Lie Groups and Lie Algebras and their Representations (Springer, New York, 1984).Google Scholar
[12]Wilson, D. C., ‘On the structure of Sidon sets’, Monatsh. Math. 101 (1986), 67–74.Google Scholar