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Lacunarity for compact groups.

R E Edwards1, E Hewitt, K A Ross

  • 1Australian National University, Canberra.

Proceedings of the National Academy of Sciences of the United States of America
|January 1, 1971
PubMed
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Researchers found a new type of mathematical set, called a [unk](q) set, within compact Abelian groups. These sets are useful for understanding trigonometric polynomials but are not Sidon sets, extending previous findings.

Area of Science:

  • Harmonic Analysis
  • Functional Analysis
  • Group Theory

Background:

  • Introduces [unk](q) sets and Sidon sets in the context of compact Abelian groups and their character groups.
  • Highlights existing knowledge that Sidon sets are a subset of [unk](q) sets.
  • References Rudin's work on the circle group (G=T) and integers (X=Z) demonstrating a set that is [unk](q) but not Sidon.

Purpose of the Study:

  • To generalize Rudin's findings to all infinite compact Abelian groups.
  • To prove the existence of a subset Delta in the character group X that is [unk](q) for all q (1 < q < infinity) but is not a Sidon set.

Main Methods:

  • Utilizes concepts from harmonic analysis and the theory of compact Abelian groups.
  • Involves the construction or identification of specific subsets within character groups.

Related Experiment Videos

  • Applies inequalities related to trigonometric polynomials and norms (parallelf parallel(q) and parallelf parallel(1)).

    • Main Results:

    • Demonstrates that for any infinite compact Abelian group G, its character group X contains a subset Delta.
    • This subset Delta satisfies the [unk](q) set property for all q (1 < q < infinity).
    • Crucially, this subset Delta is shown to not be a Sidon set.

    Conclusions:

    • Confirms the existence of [unk](q) sets that are not Sidon sets in a broader class of mathematical structures (all infinite compact Abelian groups).
    • Extends the understanding of the relationship between [unk](q) sets and Sidon sets.
    • Provides a foundation for further research into the properties and applications of these sets in harmonic analysis.