Английская Википедия:Additive basis

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In additive number theory, an additive basis is a set <math>S</math> of natural numbers with the property that, for some finite number <math>k</math>, every natural number can be expressed as a sum of <math>k</math> or fewer elements of <math>S</math>. That is, the sumset of <math>k</math> copies of <math>S</math> consists of all natural numbers. The order or degree of an additive basis is the number <math>k</math>. When the context of additive number theory is clear, an additive basis may simply be called a basis. An asymptotic additive basis is a set <math>S</math> for which all but finitely many natural numbers can be expressed as a sum of <math>k</math> or fewer elements of <math>S</math>.Шаблон:R

For example, by Lagrange's four-square theorem, the set of square numbers is an additive basis of order four, and more generally by the Fermat polygonal number theorem the polygonal numbers for <math>k</math>-sided polygons form an additive basis of order <math>k</math>. Similarly, the solutions to Waring's problem imply that the <math>k</math>th powers are an additive basis, although their order is more than <math>k</math>. By Vinogradov's theorem, the prime numbers are an asymptotic additive basis of order at most four, and Goldbach's conjecture would imply that their order is three.Шаблон:R

The unproven Erdős–Turán conjecture on additive bases states that, for any additive basis of order <math>k</math>, the number of representations of the number <math>n</math> as a sum of <math>k</math> elements of the basis tends to infinity in the limit as <math>n</math> goes to infinity. (More precisely, the number of representations has no finite supremum.)Шаблон:R The related Erdős–Fuchs theorem states that the number of representations cannot be close to a linear function.Шаблон:R The Erdős–Tetali theorem states that, for every <math>k</math>, there exists an additive basis of order <math>k</math> whose number of representations of each <math>n</math> is <math>\Theta(\log n)</math>.Шаблон:R

A theorem of Lev Schnirelmann states that any sequence with positive Schnirelmann density is an additive basis. This follows from a stronger theorem of Henry Mann according to which the Schnirelmann density of a sum of two sequences is at least the sum of their Schnirelmann densities, unless their sum consists of all natural numbers. Thus, any sequence of Schnirelmann density <math>\varepsilon > 0</math> is an additive basis of order at most <math>\lceil 1/\varepsilon\rceil</math>.Шаблон:R