# Schnirelmann density

**Schnirelmann density** is a notion of density for sets of integers, introduced in 1930 by [Lev Schnirelmann](https://www.edgepedia.org/lev-schnirelmann) (Лев Генрихович Шнирельман), defined as the infimum, over all n, of the proportion of the first n positive integers that belong to the set. Unlike asymptotic density, which describes the behavior of a set only at infinity, Schnirelmann density is sensitive to the smallest elements of a set, and this sensitivity is what makes it effective for additive questions such as the Goldbach problem.<sup>[1](https://encyclopediaofmath.org/wiki/Density_of_a_sequence)</sup>

| Key fact | Detail |
|---|---|
| Definition | For a set \( A \subseteq \mathbb{N} \) with counting function \( A(n) = |A \cap [1, n]| \), the Schnirelmann density is \( \sigma(A) = \inf_{n \ge 1} A(n)/n \)<sup>[1](https://encyclopediaofmath.org/wiki/Density_of_a_sequence)</sup> |
| Introduced | 1930, by L. G. Shnirel'man<sup>[1](https://encyclopediaofmath.org/wiki/Density_of_a_sequence)</sup> |
| Extremal value | \( \sigma(A) = 1 \) if and only if \( A \) coincides with the set of all non-negative integers<sup>[1](https://encyclopediaofmath.org/wiki/Density_of_a_sequence)</sup> |
| Sensitivity | \( \sigma(A) = 0 \) whenever \( 1 \notin A \); positive Schnirelmann density requires \( 1 \in A \) and positive lower asymptotic density<sup>[2](https://www.math.cmu.edu/users/af1p/Teaching/AdditiveCombinatorics/Additive-Combinatorics.pdf)</sup> |
| Schnirelmann's inequality | If \( \sigma(A) = \alpha \) and \( \sigma(B) = \beta \) with at least one of \( A, B \) containing 0, then \( \sigma(A+B) \ge \alpha + \beta - \alpha\beta \)<sup>[3](https://publi.math.unideb.hu/load_doc.php?p=479&t=pap)</sup> |
| Mann's theorem | If \( 0 \in A \cap B \), then \( \sigma(A+B) \ge \min(\alpha + \beta, 1) \)<sup>[3](https://publi.math.unideb.hu/load_doc.php?p=479&t=pap)</sup> |
| Goldbach application | Schnirelmann proved every integer is a sum of at most 800,000 primes, the first unconditional result of its kind<sup>[4](https://www.math.stonybrook.edu/~moira/mat331-spr10/papers/1958%20MirskyAdditive%20Prime%20Number%20Theory.pdf)</sup> |

## Definition and basic properties

Let \( A \) be a set of non-negative integers containing 0, and let \( A(n) \) count the elements of \( A \) up to \( n \). The Schnirelmann density is

\[ \sigma(A) = \inf_{n \ge 1} \frac{A(n)}{n}. \]

Some authors use the equivalent counting convention \( \sigma(A) = \inf_{n \in \mathbb{N}} |A \cap [0, n]|/(n+1) \), which is needed for statements such as Mann's theorem but leaves the results of interest unchanged.<sup>[5](https://www.combinatorics.org/ojs/index.php/eljc/article/download/v24i3p34/pdf/)</sup>

The infimum, rather than a limit, is the defining feature. Asymptotic density is translation invariant and unchanged by adding or removing finitely many elements; Schnirelmann density has neither property. In particular, \( \sigma(A) = 0 \) if \( 1 \notin A \), because the ratio \( A(1)/1 \) is then zero, and \( \sigma(A) > 0 \) holds exactly when \( 1 \in A \) and the lower asymptotic density of \( A \) is positive.<sup>[2](https://www.math.cmu.edu/users/af1p/Teaching/AdditiveCombinatorics/Additive-Combinatorics.pdf)</sup> The density reaches its maximum value only for the full set: \( \sigma(A) = 1 \) if and only if \( A \) is the entire set of non-negative integers.<sup>[1](https://encyclopediaofmath.org/wiki/Density_of_a_sequence)</sup>

## Bases and Schnirelmann's theorem

The arithmetic sum \( A + B = \{a + b : a \in A,\ b \in B\} \) is taken without repetition, and \( hA \) denotes the h-fold sumset \( A + \cdots + A \). A set \( A \) with \( hA = \mathbb{N}_0 \) is called a basis of order h; if every sufficiently large integer lies in \( hA \), \( A \) is an asymptotic basis of order h.<sup>[1](https://encyclopediaofmath.org/wiki/Density_of_a_sequence)</sup><sup> • </sup><sup>[2](https://www.math.cmu.edu/users/af1p/Teaching/AdditiveCombinatorics/Additive-Combinatorics.pdf)</sup>

**The central theorem.** Schnirelmann showed that if \( 0 \in A \) and \( \sigma(A) > 0 \), then \( A \) is a basis: there is a number \( l \), depending on the density \( \delta = \sigma(A) \), such that every integer is a sum of at most \( l \) elements of \( A \).<sup>[2](https://www.math.cmu.edu/users/af1p/Teaching/AdditiveCombinatorics/Additive-Combinatorics.pdf)</sup><sup> • </sup><sup>[4](https://www.math.stonybrook.edu/~moira/mat331-spr10/papers/1958%20MirskyAdditive%20Prime%20Number%20Theory.pdf)</sup> The proof rests on an inequality for sumsets: for sets \( A \) and \( B \) of non-negative integers with positive Schnirelmann densities \( \alpha \) and \( \beta \), if at least one of them contains 0, then

\[ \sigma(A + B) \ge \alpha + \beta - \alpha\beta. \]

Under the stronger assumption \( 0 \in A \cap B \), Mann strengthened this to \( \sigma(A+B) \ge \min(\alpha + \beta, 1) \).<sup>[3](https://publi.math.unideb.hu/load_doc.php?p=479&t=pap)</sup> Later work determined the best possible bound in Schnirelmann's inequality and described all cases of equality.<sup>[3](https://publi.math.unideb.hu/load_doc.php?p=479&t=pap)</sup>

## Application to the Goldbach problem

Schnirelmann's motivation was [Goldbach's conjecture](https://www.edgepedia.org/goldbachs-conjecture), the assertion that every integer greater than 3 is a sum of two or three primes depending on parity. His strategy had two steps. First, using sieve methods, he established that the integers representable as a sum of two primes have positive density, in the exact form \( d(2P) > 0 \) for \( P \) the set of primes; today, since almost all even integers are known to be sums of two primes, this density is known to equal \( 1/2 \).<sup>[2](https://www.math.cmu.edu/users/af1p/Teaching/AdditiveCombinatorics/Additive-Combinatorics.pdf)</sup> Second, his theorem converts positive density into a basis property: every integer is a sum of a bounded number of primes.<sup>[1](https://encyclopediaofmath.org/wiki/Density_of_a_sequence)</sup><sup> • </sup><sup>[6](https://encyclopediaofmath.org/wiki/Additive_number_theory)</sup>

The bound that emerged from Schnirelmann's analysis was \( c_6 = 800{,}000 \) primes. Although large, the result was a major step at the time, when only hypothetical results of Hardy and Littlewood were available. Subsequent writers improved the quantitative aspect; it was shown that every sufficiently large number is a sum of at most 67 primes, a result soon superseded by Vinogradov's theorems.<sup>[4](https://www.math.stonybrook.edu/~moira/mat331-spr10/papers/1958%20MirskyAdditive%20Prime%20Number%20Theory.pdf)</sup> The German literature calls the result the Satz von Goldbach-Schnirelmann, and records that Schnirelmann proved it for \( k = 800{,}000 \) as the first progress on the Goldbach problems.<sup>[7](https://www.spektrum.de/lexikon/mathematik/schnirelmann-satz-von/8819)</sup>

## Essential components and Plünnecke's inequality

A set \( B \) is an essential component if \( \sigma(A + B) > \sigma(A) \) for every \( A \) with \( 0 < \sigma(A) < 1 \). By Schnirelmann's theorem, any \( B \) with \( 0 \in B \) and \( \sigma(B) > 0 \) is an essential component.<sup>[8](https://jinr.people.charleston.edu/research/plunnecke2-revised.pdf)</sup> In 1935 Erdős proved that every additive basis \( B \) of order \( h \) is an essential component, with the explicit bound

\[ \sigma(A + B) \ge \sigma(A) + \tfrac{1}{2h}\, \sigma(A)\,(1 - \sigma(A)). \]

Plünnecke improved this to \( \sigma(A + B) \ge \sigma(A)^{1 - 1/h} \), the best exponent, in the general form

\[ \sigma(A + B) \ge \sigma(A)^{1 - \frac{1}{k}} \, \sigma(kB)^{\frac{1}{k}} \]

for positive integers \( k \) and sets \( A, B \) with \( 0 \in B \).<sup>[9](https://www.sciencedirect.com/science/article/pii/S0022314X85710980)</sup><sup> • </sup><sup>[5](https://www.combinatorics.org/ojs/index.php/eljc/article/download/v24i3p34/pdf/)</sup> A simplified proof of Plünnecke's inequality for Schnirelmann density, avoiding the impact function, was later given by Renling Jin, a mathematician at the [College of Charleston](https://www.edgepedia.org/college-of-charleston) working in additive number theory.<sup>[8](https://jinr.people.charleston.edu/research/plunnecke2-revised.pdf)</sup>

The same inequality extends to other densities with qualifications. It holds for the lower asymptotic density \( d(A) = \liminf_{n \to \infty} A(n)/n \), first obtained by Imre Ruzsa with alternative proofs by Jin, and for the upper Banach density, but the corresponding statement for the upper asymptotic density is false.<sup>[5](https://www.combinatorics.org/ojs/index.php/eljc/article/download/v24i3p34/pdf/)</sup><sup> • </sup><sup>[8](https://jinr.people.charleston.edu/research/plunnecke2-revised.pdf)</sup> These extensions yield concrete consequences, such as \( d(A + P) > d(A)^{2/3} \) for any \( A \subseteq \mathbb{N} \), where \( P \) is the set of primes.<sup>[8](https://jinr.people.charleston.edu/research/plunnecke2-revised.pdf)</sup>

## Relation to other densities

Schnirelmann density sits in a family of set functions used to measure the size of sumsets, alongside asymptotic density, lower and upper asymptotic density, and Banach density. For any set \( A \), the Schnirelmann density is bounded above by the lower asymptotic density, since an infimum over all \( n \) cannot exceed any limiting value.<sup>[8](https://jinr.people.charleston.edu/research/plunnecke2-revised.pdf)</sup> Its lack of translation invariance makes it less natural as a measure of asymptotic size, but the same sensitivity to initial segments is exactly what the basis theorems require.<sup>[2](https://www.math.cmu.edu/users/af1p/Teaching/AdditiveCombinatorics/Additive-Combinatorics.pdf)</sup> Mann's theorem was generalized by Dyson in 1945, who obtained a lower bound for the Schnirelmann density of rank-\( r \) sumsets.<sup>[10](https://arxiv.org/abs/2407.12253)</sup> The concept has also been formalized in the Lean mathematical library Mathlib, which includes the basic positivity and sumset properties of Schnirelmann density.<sup>[11](https://marcodavid.net/ainfinity-lean/docs/Mathlib/Combinatorics/Schnirelmann.html)</sup>

## References

1. [Density of a sequence, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Density_of_a_sequence)
2. [Additive Combinatorics lecture notes (Florian Pop), Carnegie Mellon University](https://www.math.cmu.edu/users/af1p/Teaching/AdditiveCombinatorics/Additive-Combinatorics.pdf)
3. [P. Hegedűs, G. Piroska, I. Z. Ruzsa (1998). On the Schnirelmann density of sumsets. Publ. Math. Debrecen 53/3-4, 333–345](https://publi.math.unideb.hu/load_doc.php?p=479&t=pap)
4. [L. Mirsky (1958). Additive Prime Number Theory](https://www.math.stonybrook.edu/~moira/mat331-spr10/papers/1958%20MirskyAdditive%20Prime%20Number%20Theory.pdf)
5. [Multidimensional lower density versions of Plünnecke's inequality, Electronic Journal of Combinatorics 24(3) (2017), #P3.34](https://www.combinatorics.org/ojs/index.php/eljc/article/download/v24i3p34/pdf/)
6. [Additive number theory, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Additive_number_theory)
7. [Schnirelmann, Satz von, Lexikon der Mathematik, Spektrum Verlag](https://www.spektrum.de/lexikon/mathematik/schnirelmann-satz-von/8819)
8. [Renling Jin. Density Versions of Plünnecke Inequality – Epsilon-Delta Approach](https://jinr.people.charleston.edu/research/plunnecke2-revised.pdf)
9. [Malouf J. L. (1995). On a Theorem of Plünnecke Concerning the Sum of A Basis and A Set of Positive Density. Journal of Number Theory 54(1), 12–22](https://www.sciencedirect.com/science/article/pii/S0022314X85710980)
10. [Additive number theory and the Dyson transform, arXiv:2407.12253](https://arxiv.org/abs/2407.12253)
11. [Mathlib.Combinatorics.Schnirelmann documentation](https://marcodavid.net/ainfinity-lean/docs/Mathlib/Combinatorics/Schnirelmann.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Analytic number theory › Additive number theory › Schnirelmann density and additive density methods*

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