## Abstract

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 function s, then the zero-one random variables s(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.

Original language | English |
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Pages (from-to) | 523-531 |

Number of pages | 9 |

Journal | Probability Theory and Related Fields |

Volume | 76 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Dec 1987 |