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.
[1][2] 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.
[3] 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.
, then Lagrange's four-square theorem can be restated as
, and one might ask at what point the sumset attains Schnirelmann density 1 and how does it increase.
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.
The theorem provides the first insights on how sumsets accumulate.
It seems unfortunate that its conclusion stops short of showing
Yet, Schnirelmann provided us with the following results, which sufficed for most of his purpose.
for a finite sum, is called an additive basis, and the least number of summands required is called the degree (sometimes order) of the basis.
Thus, the last theorem states that any set with positive Schnirelmann density is an additive basis.
Historically the theorems above were pointers to the following result, at one time known as the
It was used by Edmund Landau and was finally proved by Henry Mann in 1942.
An analogue of this theorem for lower asymptotic density was obtained by Kneser.
[4] At a later date, E. Artin and P. Scherk simplified the proof of Mann's theorem.
to be the number of non-negative integral solutions to the equation and
to be the number of non-negative integral solutions to the inequality in the variables
The hard part is to show that this bound still works on the average, i.e., Lemma.
We have thus established the general solution to Waring's Problem: Corollary.
In 1930 Schnirelmann used these ideas in conjunction with the Brun sieve to prove Schnirelmann's theorem,[1][2] that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant:[6] Schnirelmann obtained C < 800000.
[7] Schnirelmann's constant is the lowest number C with this property.
[6] Olivier Ramaré showed in (Ramaré 1995) that Schnirelmann's constant is at most 7,[6] improving the earlier upper bound of 19 obtained by Hans Riesel and R. C. Vaughan.
[6] In 2013, Harald Helfgott proved Goldbach's weak conjecture for all odd numbers.
[8][9][10][11] Khintchin proved that the sequence of squares, though of zero Schnirelmann density, when added to a sequence of Schnirelmann density between 0 and 1, increases the density: This was soon simplified and extended by Erdős, who showed, that if A is any sequence with Schnirelmann density α and B is an additive basis of order k then and this was improved by Plünnecke to Sequences with this property, of increasing density less than one by addition, were named essential components by Khintchin.
Linnik showed that an essential component need not be an additive basis[14] as he constructed an essential component that has xo(1) elements less than x.
This was improved by E. Wirsing to For a while, it remained an open problem how many elements an essential component must have.
Finally, Ruzsa determined that for every ε > 0 there is an essential component which has at most c(log x)1+ε elements up to x, but there is no essential component which has c(log x)1+o(1) elements up to x.