6 - Probabilistic sieves

Summary

The content of this chapter is a kind of warm-up to the next. Both involve applications of sieves where the siftable set is a general measure space (X, μ), not simply a finite set with counting measure. The results described in this chapter may well be amenable to other proofs based on classical sieves, but this will not be the case in the next chapter. Moreover, alternative proofs may not be always possible if we go further along the route we describe …

The idea we want to pursue is to work with a given sieve setting (such as Ψ = (Z, {primes}, Z → Z/ℓZ)), using siftable sets which are probability spaces, given with a Y-valued random variable. Then we may look at the probability that the random variable lies in some sifted subset of Y, and as usual this may give information on the probability that the random variable satisfies certain properties which may be described or approached with sieve conditions. We pursue this in two ‘abelian’ cases here, before looking at non-abelian groups in the next chapter.

Probabilistic sieves with integers

Our first example is the analogue of the classical sieve of intervals of integers. Consider a probability space (Ω, Σ, P) (i.e., P is a probability measure on Ω, which should not be confused with the sieving sets Ω_ℓ, with respect to a σ-algebra Σ; see Appendix F for a survey of probabilistic language, for readers unfamiliar with it), and let F = N : Ω → Z be an integer-valued random variable.