Schnirelmann density

Summary

The Schnirelmann density of a set $A$ of positive integers is the infimum over $n \ge 1$ of the proportion of $\set{1, \dots, n}$ that lies in $A$. Unlike the natural density, it always exists and it is very sensitive to small elements: it is $0$ whenever $1 \notin A$. It behaves well under sums of sets, which makes it a tool for additive problems. Schnirelmann used it to show that every integer greater than $1$ is a sum of a bounded number of primes.