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.
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Cameron, P.J. On the structure of a random sum-free set. Probab. Th. Rel. Fields 76, 523–531 (1987). https://doi.org/10.1007/BF00960072
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DOI: https://doi.org/10.1007/BF00960072