The Schnirelmann density of a set of natural numbersA is defined as where A 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 as fails to exist.
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.
Schnirelmann's theorems
If we set, then Lagrange's four-square theorem can be restated as. 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 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.
A subset with the property that for a finite sum, is called an additive basis, and the least number of summands required is called the degree of the basis. Thus, the last theorem states that any set with positive Schnirelmann density is an additive basis. In this terminology, the set of squares is an additive basis of degree 4.
Mann's theorem
Historically the theorems above were pointers to the following result, at one time known as the hypothesis. It was used by Edmund Landau and was finally proved by Henry Mann in 1942.
Theorem. Let and be subsets of. In case that, we still have
An analogue of this theorem for lower asymptotic density was obtained by Kneser. At a later date, E. Artin and P. Scherk simplified the proof of Mann's theorem.
Waring's problem
Let and be natural numbers. Let. Define 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, respectively. Thus. We have The volume of the -dimensional body defined by, is bounded by the volume of the hypercube of size, hence. The hard part is to show that this bound still works on the average, i.e.,
Lemma. For all there exists and a constant, depending only on, such that for all,
for all
With this at hand, the following theorem can be elegantly proved.
Theorem. For all there exists for which.
We have thus established the general solution to Waring's Problem:
Corollary. For all there exists, depending only on, such that every positive integer can be expressed as the sum of at most many -th powers.
Schnirelmann's constant
In 1930 Schnirelmann used these ideas in conjunction with the Brun sieve to prove Schnirelmann's theorem, that any natural numbergreater than 1 can be written as the sum of not more than Cprime numbers, where C is an effectively computable constant: Schnirelmann obtained C < 800000. Schnirelmann's constant is the lowest number C with this property. Olivier Ramaré showed in that Schnirelmann's constant is at most 7, improving the earlier upper bound of 19 obtained by Hans Riesel and R. C. Vaughan. Schnirelmann's constant is at least 3; Goldbach's conjecture implies that this is the constant's actual value. In 2013, Harald Helfgott proved Goldbach's weak conjecture for all odd numbers. Therefore Schnirelmann's constant is at most 4.
Essential components
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 as he constructed an essential component that has xo elements less than x. More precisely, the sequence has elements less than x for some c < 1. 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 an essential component has at least c elements up to x, for some c > 1, and for every c > 1 there is an essential component which has at most c elements up to x.
