Schnirelmann Density - Schnirelmann's Theorems

Schnirelmann's Theorems

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 . If then there exists such that