Additive number theory

In additive number theory , the behavior of sets of whole numbers is examined under addition. In multiplicative number theory, on the other hand, the focus is on behavior under multiplication (theory of prime numbers ).

For problems of additive number theory, as in multiplicative number theory, methods of analytical number theory were often used, such as the Hardy-Littlewood circle method and the method of trigonometric sums by Ivan Matwejewitsch Vinogradow as well as sieving methods . An essay by Lew Genrichowitsch Schnirelman from 1930, in which he introduced densities of sets of natural numbers (Schnirelmann density) and thus made statements about the sums of sets of natural numbers and results in the context of, was of particular influence on the orientation of this area of ​​number theory Goldbach's conjecture found. If a subset of natural numbers, then the Schnirelmann density is defined as: ${\displaystyle A}$

${\displaystyle \sigma (A)=\inf _{n\in \mathbb {N} }{\frac {A(n)}{n}}}$

with the number of elements less than or equal to . The sum set (Minkowski sum) is the set of all sums with and . Correspondingly stands for the k times the sum . For example, the set of prime numbers comes into consideration as sets . ${\displaystyle A(n)}$ ${\displaystyle A}$ ${\displaystyle n}$ ${\displaystyle A+B}$ ${\displaystyle a+b}$ ${\displaystyle a\in A}$ ${\displaystyle b\in B}$ ${\displaystyle kA}$ ${\displaystyle A+\cdots +A}$ ${\displaystyle P}$

Results and Problems

The following are some of the important results and problems of additive number theory:

• Which natural numbers can be represented as the sum of two squares? This can be traced back to the question already answered by Pierre de Fermat and Leonhard Euler ( two-squares theorem ), which prime numbers can be represented as the sum of two squares. ${\displaystyle (a^{2}+b^{2})(c^{2}+d^{2})={(ac-bd)}^{2}+{(ad+bc)}^{2}}$
• Four squares theorem by Joseph Louis Lagrange
• The Goldbach hypothesis and results from its environment. The guess is that the even number contains greater than 2. ${\displaystyle 2P}$
• The Waring problem : How large must in be for this amount contains all natural numbers? It is . ${\displaystyle h}$ ${\displaystyle hA_{k}}$ ${\displaystyle A_{k}=\lbrace 0^{k},1^{k},2^{k},3^{k},\cdots \rbrace }$
• Proof of the ( ) hypothesis by Schnirelmann and Edmund Landau by Henry Mann , who received the Cole Prize for it : Be , then applies . ${\displaystyle \alpha +\beta }$ ${\displaystyle A+B\neq \mathbb {N} }$ ${\displaystyle \sigma (A+B)\geq \sigma (A)+\sigma (B)}$

Melvyn Nathanson distinguishes between direct problems of classical additive number theory, which are about determining the structure and properties of sum sets . In addition, there are inverse problems in which it is a matter of obtaining statements about the structure of and from the structure of the sum set . A prototypical example is Gregory Freiman's theorem . be a finite set of natural numbers and the sum set small (that is, there is a constant with ). Then there is a n-dimensional arithmetic sequence of length which contains, where the constant only by and dependent. ${\displaystyle kA}$ ${\displaystyle A+B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle 2A}$ ${\displaystyle c}$ ${\displaystyle |A+A|<c|A|}$ ${\displaystyle c_{1}|A|}$ ${\displaystyle A}$ ${\displaystyle c_{1}}$ ${\displaystyle c}$ ${\displaystyle n}$

There are connections to additive combinatorics. Sometimes problems of Diophantine geometry were also included, but methods of their own are used here.

Instead of adding in natural numbers, one can also look at general Abelian groups or semigroups.

literature

• Melvyn B. Nathanson: Additive Number Theory , 2 volumes (Volume 1: The Classical Bases, Volume 2: Inverse Problems and the Geometry of Sumsets), Graduate Texts in Mathematics 164/165, Springer-Verlag 1996.

Web links

• BM Bredikhin, Additive Number Theory , Encyclopedia of Mathematics, Springer

Individual evidence

1. ↑ Schnirelmann, About additive properties of numbers, Mathematische Annalen. Volume 107, 1933, pp. 649-690
2. ^ Mann: A proof of the fundamental theorem on the density of sums of sets of positive integers. Annals of Mathematics, Series 2, Vol. 43, 1942, pp. 523-527. The problem and the solution of man be in Khinchin: Three pearls of number theory treated
3. ^ Nathanson, Additive Number Theory, 2 volumes, Springer 1996. He deals with inverse problems in the second volume
4. ^ Freiman, Foundation of a structural theory of set addition, AMS Translations of Mathematical Monographs 37, 1973, first in Russian 1966
5. ^ Nathanson, Additive Number Theory, 1996, Volume 2. There, Freiman's theorem is a central theme
6. ^ Terence Tao , Van H. Vu : Additive Combinatorics, Cambridge UP 2006
7. ↑ For example, Hans-Heinrich Ostmann counts the Fermat problem in his volume Additive Number Theory , 2 volumes, Results of Mathematics and their Border Areas , Springer 1956, 1957 (here Volume 2)
8. ^ Henry Mann, Addition Theorems: The Addition Theorems of Group Theory and Number Theory, Krieger 1976