Mann's theorem

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 L.G. 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 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 as we shall see.

If we set ${\displaystyle {\mathfrak {G}}^{2}=\{k^{2}\}_{k=1}^{\infty }}$, then Lagrange's four-square theorem can be restated as ${\displaystyle \sigma ({\mathfrak {G}}^{2}\oplus {\mathfrak {G}}^{2}\oplus {\mathfrak {G}}^{2}\oplus {\mathfrak {G}}^{2})=1}$. (Here the symbol ${\displaystyle A\oplus B}$ denotes the sumset of ${\displaystyle A\cup \{0\}}$ and ${\displaystyle B\cup \{0\}}$.) It is clear that ${\displaystyle \sigma {\mathfrak {G}}^{2}=0}$. In fact, we still have ${\displaystyle \sigma ({\mathfrak {G}}^{2}\oplus {\mathfrak {G}}^{2})=0}$, and one might ask at what point the sumset attains Schnirelmann density 1 and how does it increase. It actually is the case that ${\displaystyle \sigma ({\mathfrak {G}}^{2}\oplus {\mathfrak {G}}^{2}\oplus {\mathfrak {G}}^{2})=5/6}$ and one sees that sumsetting ${\displaystyle {\mathfrak {G}}^{2}}$ once again yields a more populous set, namely all of ${\displaystyle \mathbb {N} }$. 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.

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