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Probability Theory and Related Fields

, Volume 76, Issue 4, pp 523–531 | Cite as

On the structure of a random sum-free set

  • Peter J. Cameron
Article

Summary

There is a natural probability measure on the set Σ of all sum-free sets of natural numbers. If we represent such a set by its characteristic functions, then the zero-one random variabless(i) are far from independent, and we cannot expect a law of large numbers to hold for them. In this paper I conjecture a decomposition of Σ into countably many more tractible pieces (up to a null set). I prove that each piece has positive measure, and show that, within each piece, a random set almost surely has a density which is a fixed rational number depending only on the piece. For example, the first such piece is made up of sets consisting entirely of odd numbers; it has probability 0.218 ..., and its members almost surely have density 1/4.

Keywords

Stochastic Process Probability Measure Characteristic Function Probability Theory Natural Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Peter J. Cameron
    • 1
  1. 1.School of Mathematical SciencesQueen Mary CollegeLondonUK

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