The probabilistic method

Abstract

In additive number theory, one frequently faces the problem of showing that a set A contains a subset B with a certain property P . A very powerful tool for such a problem is Erdős’ probabilistic method. In order to show that such a subset B exists, it suffices to prove that a properly defined random subset of A satisfies P with positive probability. The power of the probabilistic method has been justified by the fact that in most problems solved using this approach, it seems impossible to come up with a deterministically constructive proof of comparable simplicity. In this chapter we are going to present several basic probabilistic tools together with some representative applications of the probabilistic method, particularly with regard to additive bases and the primes. We shall require several standard facts about the distribution of primes P = {2, 3, 5, …}; so as not to disrupt the flow of the chapter we have placed these facts in an appendix (Section 1.10). Notation. We assume the existence of some sample space (usually this will be finite). If E is an event in this sample space, we use P ( E ) to denote the probability of E , and I ( E ) to denote the indicator function (thus I ( E ) = 1 if E occurs and 0 otherwise). If E, F are events, we use E ∧ F to denote the event that E, F both hold, E ∨ F to denote the event that at least one of E, F hold, and Ē to denote the event that E does not hold.