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


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.


Stochastic Process Probability Measure Characteristic Function Probability Theory Natural Number 
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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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