Spacings between integers having typically many prime factors

Abstract

We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\de$ arbitrarily small and positive, the nearest neighbor spacings between integers $n$ with $|\om(n)-\log_2 n|\le (\log_2 n)^{\de}$ obey the Poisson distribution law.