## Lacunary sets based on Lorentz spaces

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- by Raymond J. Grinnell PDF
- Proc. Amer. Math. Soc.
**127**(1999), 3547-3556 Request permission

## 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.## References

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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
- Email: grinnell@uwichill.edu.bb
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**127**(1999), 3547-3556 - MSC (1991): Primary 43A46; Secondary 43A15, 43A25
- DOI: https://doi.org/10.1090/S0002-9939-99-04918-7
- MathSciNet review: 1610901