Schnirelmann density

In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it. The Schnirelmann density of a set of natural numbers A is defined as where A(n) denotes the number of elements of A not exceeding n and inf is infimum. The Schnirelmann density is well-defined even if the limit of A(n)/n as n → ∞ fails to exist (see upper and lower asymptotic density). By definition, 0 ≤ A(n) ≤ n and n σA ≤ A(n) for all n, and therefore 0 ≤ σA ≤ 1, and σA = 1 if and only if A = N. Furthermore, The Schnirelmann density is sensitive to the first values of a set: In particular, and Consequently, the Schnirelmann densities of the even numbers and the odd numbers, which one might expect to agree, are 0 and 1/2 respectively. Schnirelmann and Yuri Linnik exploited this sensitivity. If we set , then Lagrange's four-square theorem can be restated as . (Here the symbol denotes the sumset of and .) It is clear that . In fact, we still have , and one might ask at what point the sumset attains Schnirelmann density 1 and how does it increase. It actually is the case that and one sees that sumsetting once again yields a more populous set, namely all of . Schnirelmann further succeeded in developing these ideas into the following theorems, aiming towards Additive Number Theory, and proving them to be a novel resource (if not greatly powerful) to attack important problems, such as Waring's problem and Goldbach's conjecture. Theorem. Let and be subsets of . Then Note that . Inductively, we have the following generalization. Corollary. Let be a finite family of subsets of . Then The theorem provides the first insights on how sumsets accumulate. It seems unfortunate that its conclusion stops short of showing being superadditive. Yet, Schnirelmann provided us with the following results, which sufficed for most of his purpose. Theorem. Let and be subsets of . If , then Theorem. (Schnirelmann) Let .