On the structure of a random sum-free set of positive integers

Cameron introduced a natural probability measure on the set S of sum-free sets, and asked which sets of sum-free sets have a positive probability of occurring in this probability measure. He showed that the set of subsets of the odd numbers has a positive probability, and that the set of subsets of any sum-free set corresponding to a complete modular sum-free set also has a positive probability of occurring. In this paper we consider, for every sumfree set S, the representation function rS(n), and show that if rS(n) grows sufficiently quickly then the set of subsets of S has positive probability, and conversely, that if rS(n) has a sub-sequence with suitably slow growth, then the set of subsets of S has probability zero. The results include those of Cameron mentioned above as particular cases.