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Lacunary sets based on Lorentz spaces

Author: Raymond J. Grinnell
Journal: Proc. Amer. Math. Soc. 127 (1999), 3547-3556
MSC (1991): Primary 43A46; Secondary 43A15, 43A25
Published electronically: May 13, 1999
MathSciNet review: 1610901
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Abstract: A new lacunary set for compact abelian groups is introduced; this is called a $\Lambda(p,q)$ set. This set is defined in terms of the Lorentz spaces and is shown to be a generalization of $\Lambda(p)$ sets and Sidon sets. A number of functional-analytic statements about $\Lambda(p,q)$ sets are established by making use of the structural similarities between $L^{p}$ spaces and Lorentz spaces. These statements are analogous to several well-known properties of a set which are equivalent to the definition of a $\Lambda(p)$ set. Some general set-theoretic and arithmetic properties of $\Lambda(p,q)$ sets are also developed; these properties extend known results on the structure of $\Lambda(p)$ sets. Open problems and directions for further research are outlined.

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Additional Information

Raymond J. Grinnell
Affiliation: Department of Computer Science, Mathematics & Physics, University of the West Indies, Cave Hill Campus, P.O. Box 64, Bridgetown, Barbados, West Indies

Received by editor(s): September 5, 1996
Received by editor(s) in revised form: February 12, 1998
Published electronically: May 13, 1999
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1999 American Mathematical Society

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