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The analytic content is not semiadditive.

Eduardo S Zeron1, Paul M Gauthier2

  • 1Departamento de Matemáticas, Cinvestav del IPN, Apartado Postal 14-740, 07000 Ciudad de México, CDMX México.

Analysis and Mathematical Physics
|December 2, 2024
PubMed
Summary
This summary is machine-generated.

Analytic content, a measure of complex set properties, is not subadditive or semiadditive. Compact sets with positive analytic content cannot be formed by countable unions of zero-content sets.

Keywords:
Analytic contentRational functionsUniform approximation

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Area of Science:

  • Complex Analysis
  • Set Theory

Background:

  • The analytic content of a compact set K in the complex plane quantifies its relation to rational functions.
  • Understanding the additivity properties of analytic content is crucial for complex analysis and geometry.

Purpose of the Study:

  • To investigate the additivity properties (subadditivity and semiadditivity) of analytic content for compact sets in the complex plane.
  • To explore the decomposition of compact sets based on their analytic content.

Main Methods:

  • Defining analytic content as the K-uniform distance from complex conjugation to the algebra of rational functions with poles outside K.
  • Proving that any compactum K can be decomposed into two new compact sets K1 and K2 such that the analytic content of K equals the maximum of the analytic contents of K1 and K2.
  • Demonstrating that a compactum with positive analytic content cannot be a countable union of sets with zero analytic content.

Main Results:

  • The analytic content is proven to be neither subadditive nor semiadditive.
  • A compact set K can be decomposed into K1 U K2 such that analytic_content(K) = max(analytic_content(K1), analytic_content(K2)).
  • Compact sets with positive analytic content cannot be represented as a countable union of sets with zero analytic content.

Conclusions:

  • The analytic content exhibits non-additive behavior, challenging previous assumptions.
  • The decomposition property provides new insights into the structure of compact sets in the complex plane.
  • Results have implications for understanding the capacity and structure of sets in complex analysis.